NCERT Solutions Class 11 Mathematics Chapter 11 Conic Sections

Exercise 11.1

 

1.  Find the equation of the circle with centre (0, 2) and radius 2.
Solution
The equation of a circle with centre (h, k) and radius r is given as
(x­ – h)² + (y ­– k)² = r²
It is given that centre (h, k) = (0, 2) and radius (r) = 2.
Therefore, the equation of the circle is
(x – 0)² + (y – 2)² = 2²
⇒ x² + y² + 4 ­– 4y = 4
⇒ x² + y² ­– 4y = 0

2. Find the equation of the circle with centre (–2, 3) and radius 4.
Solution
The equation of a circle with centre (h, k) and radius r is given as
(x­ – h)² + (y ­– k)² = r²
It is given that centre (h, k) = (–2, 3) and radius (r) = 4.
Therefore, the equation of the circle is
(x + 2)² + (y – 3)² = (4)²
⇒ x² + 4x + 4 + y² – 6y + 9 = 16
⇒ x² + y² + 4x – 6y – 3 = 0

3. Find the equation of the circle with (1/2, 1/4)and radius 1/12 . 
Solution
The equation of a circle with centre (h, k) and radius r is given as
(x­ – h)² + (y ­– k)² = r²
It is given that centre (h, k) = (1/2, 1/4) and radius (r) = (1/12).  
Therefore, the equation of the circle is

⇒ 144x² - 144x + 36 + 144y² - 72y + 9 - 1 = 0 
⇒ 144x² - 144x + 144y² - 72y + 44 = 0 
⇒ 36x² - 36x + 36y² - 18y + 11 = 0 
⇒ 36x² + 36y² - 36x - 18y + 11 = 0 

4. Find the equation of the circle with centre (1, 1) and radius √2 . 
Solution
The equation of a circle with centre (h, k) and radius r is given as
(x­ – h)² + (y ­– k)² = r²
It is given that centre (h, k) = (1, 1) and radius (r) = √2
Therefore, the equation of the circle is 
(x - 1)² + (y - 1)² = (√2)² 
⇒ x² - 2x + 1 - y² - 2y + 1 = 2 
⇒ x² + y² - 2x - 2y = 0 

5. Find the equation of the circle with centre (–a, –b) and radius √(a² - b²)
Solution
The equation of a circle with centre (h, k) and radius r is given as
(x­ – h)² + (y ­– k)² = r²
It is given that centre (h, k) = (–a, –b) and radius (r) =.
Therefore, the equation of the circle is 
(x + a)² + (y + b)² = [√(a² - b² )]² 
⇒ x² + 2ax + a² + y² + 2by + b² = a² - b² 
⇒ x² + y² + 2ax + 2by + 2b² = 0

6. Find the centre and radius of the circle (x + 5)² + (y – 3)² = 36
Solution
The equation of the given circle is (x + 5)² + (y – 3)² = 36.
(x + 5)² + (y – 3)² = 36
⇒ {x – (–5)}² + (y – 3)² = 6²,
which is of the form (x – h)² + (y – k)² = r², where h = –5, k = 3, and r = 6.
Thus, the centre of the given circle is (–5, 3), while its radius is 6.

7. Find the centre and radius of the circle x² + y² – 4x – 8y – 45 = 0
Solution
The equation of the given circle is x² + y² – 4x – 8y – 45 = 0.
x² + y² – 4x – 8y – 45 = 0
⇒ (x² – 4x) + (y² – 8y) = 45
⇒ {x² – 2(x)(2) + 2²} + {y² – 2(y)(4)+ 4²} – 4 –16 = 45
⇒ (x – 2)² + (y –4)² = 65
⇒ (x – 2)² + (y –4)² = (√65)²,
which is of the form (x – h)² + (y – k)² = r², where h = 2, k = 4, and r = √65.
Thus, the centre of the given circle is (2, 4), while its radius is √65.

8. Find the centre and radius of the circle x² + y² – 8x + 10y – 12 = 0
Solution
The equation of the given circle is x² + y² – 8x + 10y – 12 = 0.
x² + y² – 8x + 10y – 12 = 0
⇒ (x² – 8x) + (y² + 10y) = 12
⇒ {x² – 2(x)(4) + 4²} + {y² + 2(y)(5) + 5²}– 16 – 25 = 12
⇒ (x – 4)² + (y + 5)² = 53
⇒ (x - 4)² + [y - (-5)]² = (√53)² ,
which is of the form (x – h)² + (y – k)² = r², where h = 4, k = –5, and r = √53 .
Thus, the centre of the given circle is (4, –5), while its radius is √53.

9. Find the centre and radius of the circle 2x² + 2y² – x = 0
Solution
The equation of the given circle is 2x² + 2y² – x = 0.
2x² + 2y² - x = 0 
⇒ (2x² - x) + 2y² = 0 

which is of the form (x - h)² + (y - k)² = r², where h = 1/4 , k = 0 and r = 1/4. 
Thus, the centre of the given circle is (1/4, 0), while its radius is 1/4. 

10. Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.
Solution
Let the equation of the required circle be (x – h)² + (y – k)² = r².
Since the circle passes through points (4, 1) and (6, 5),
(4 – h)² + (1 – k)² = r²                 …(1)
(6 – h)² + (5 – k)² = r²                 …(2)
Since the centre (h, k) of the circle lies on line 4x + y = 16,
4h + k = 16                 …(3)
From equations (1) and (2), we obtain
(4 – h)² + (1 – k)² = (6 – h)² + (5 – k)²
⇒ 16 – 8h + h² + 1 – 2k + k² = 36 – 12h + h² + 25 – 10k + k²
⇒ 16 – 8h + 1 – 2k = 36 – 12h + 25 – 10k
⇒ 4h + 8k = 44
⇒ h + 2k = 11                 …(4)
On solving equations (3) and (4), we obtain h = 3 and k = 4.
On substituting the values of h and k in equation (1), we obtain
(4 – 3)² + (1 – 4)² = r²
⇒ (1)² + (– 3)² = r²
⇒ 1 + 9 = r²
⇒ r² = 10
⇒ r = √10
Thus, the equation of the required circle is
(x – 3)² + (y – 4)² = (√10)² 
x² – 6x + 9 + y² ­– 8y + 16 = 10
x² + y² – 6x – 8y + 15 = 0

11. Find the equation of the circle passing through the points (2, 3) and (–1, 1) and whose centre is on the line x – 3y – 11 = 0.
Solution
Let the equation of the required circle be (x – h)² + (y – k)² = r².
Since the circle passes through points (2, 3) and (–1, 1),
(2 – h)² + (3 – k)² = r²                 …(1)
(–1 – h)² + (1 – k)² = r²                 …(2)
Since the centre (h, k) of the circle lies on line x – 3y – 11 = 0,
h – 3k = 11                 …(3)
From equations (1) and (2), we obtain
(2 – h)² + (3 – k)² = (–1 – h)² + (1 – k)²
⇒ 4 – 4h + h² + 9 – 6k + k² = 1 + 2h + h² + 1 – 2k + k²
⇒ 4 – 4h + 9 – 6k = 1 + 2h + 1 – 2k
⇒ 6h + 4k = 11                 …(4)
On solving equations (3) and (4), we obtain h = 7/2 and k = -5/2. 
On substituting the values of h and k in equation (1), we obtain

4x² - 28x + 49 + 4y² + 20y + 25 = 130 
⇒ 4x² + 4y² - 28x + 20y - 56 = 0 
⇒ 4(x² + y² - 7x +5y - 14) = 0
⇒ x² + y² - 7x + 5y - 14 = 0

12. Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2, 3).
Solution 
Let the equation of the required circle be (x – h)² + (y – k)² = r².
Since the radius of the circle is 5 and its centre lies on the x-axis, k = 0 and r = 5.
Now, the equation of the circle becomes (x – h)² + y² = 25.
It is given that the circle passes through point (2, 3).
∴ (2 - h)² + 3² = 25 
⇒ (2 - h)² = 25 - 9 
⇒ (2 - h)² = 16 
⇒ 2 - h = ± √16 = ± 4
If 2 - h = 4,  then h = -2. 
If 2 - h = -4, then h = 6. 
When h = -2, the equation of the circle becomes
(x + 2)² + y² = 25
⇒ x² + 4x + 4 + y² = 25
⇒ x² + y² + 4x – 21 = 0
When h = 6, the equation of the circle becomes
(x – 6)² + y² = 25
⇒ x² – 12x +36 + y² = 25
⇒ x² + y² – 12x + 11 = 0

13. Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.
Solution
Let the equation of the required circle be (x – h)² + (y – k)² = r².
Since the circle passes through (0, 0),
(0 – h)² + (0 – k)² = r²
⇒ h² + k² = r²
The equation of the circle now becomes (x – h)² + (y – k)² = h² + k².
It is given that the circle makes intercepts a and b on the coordinate axes. This means that the circle passes through points (a, 0) and (0, b). Therefore,
(a – h)² + (0 – k)² = h² + k²                 …(1)
(0 – h)² + (b – k)² = h² + k²                 …(2)
From equation (1), we obtain
a² – 2ah + h² + k² = h² + k²
⇒ a² – 2ah = 0
⇒ a(a – 2h) = 0
⇒ a = 0 or (a – 2h) = 0
However, a ≠ 0; hence, (a – 2h) = 0 ⇒ h = a/2 .
From equation (2), we obtain
h² + b² – 2bk + k² = h² + k²
⇒ b² – 2bk = 0
⇒ b(b – 2k) = 0
⇒ b = 0 or(b – 2k) = 0
However, b ≠ 0; hence, (b – 2k) = 0 ⇒ k = b/2.
Thus, the equation of the required circle is

⇒ 4x² - 4ax + a² + 4y² - 4by + b² = a²  + b² 
⇒ 4x² + 4y² - 4ax - 4by = 0 
⇒ x² + y² - ax - by = 0

14. Find the equation of a circle with centre (2, 2) and passes through the point (4, 5).
Solution
The centre of the circle is given as (h, k) = (2, 2).
Since the circle passes through point (4, 5), the radius (r) of the circle is the distance between the points (2, 2) and (4, 5).

Thus, the equation of the circle is 
(x - h)² + (y - k)²  = r² 
⇒ (x - 2)² + (y - 2)² = (√13)² 
⇒ x² - 4x + 4 + y² - 4y + 4 = 13
⇒ x² + y² - 4x - 4y - 5 = 0

15. Does the point (–2.5, 3.5) lie inside, outside or on the circle x² + y² = 25?
Solution
The equation of the given circle is x² + y² = 25.
x² + y² = 25
⇒ (x – 0)² + (y – 0)² = 5², which is of the form (x – h)² + (y – k)² = r², where h = 0, k = 0, and r = 5.
∴ Centre = (0, 0) and radius = 5
Distance between point (–2.5, 3.5) and centre (0, 0)

= 4.3 (approx.) < 5 
Since the distance between point (-2.5, 3.5) and centre (0, 0) of the circle is less than the radius of the circle, point (–2.5, 3.5) lies inside the circle.

Exercise 11.2

1. Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y² = 12x
Solution
The given equation is y² = 12x.
Here, the coefficient of x is positive. Hence, the parabola opens towards the right.
On comparing this equation with y² = 4ax, we obtain
4a = 12
⇒ a = 3
∴Coordinates of the focus = (a, 0) = (3, 0)
Since the given equation involves y², the axis of the parabola is the x-axis.
Equation of direcctrix, x = –a i.e., x = – 3 i.e., x + 3 = 0
Length of latus rectum = 4a = 4 × 3 = 12

2. Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = 6y
Solution
The given equation is x² = 6y.
Here, the coefficient of y is positive. Hence, the parabola opens upwards.
On comparing this equation with x² = 4ay, we obtain 
4a = 6
⇒ a = 3/2 
∴ Coordinates of the focus = (0, a) = (0, 3/2) 
Since the given equation involves x² , the axis of the parabola is the y - axis.  
Equation of directrix , y = -a i.e., y = -3/2 
Length of latus rectum  = 4a = 6

3. Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y² = – 8x
Solution
The given equation is y² = –8x.
Here, the coefficient of x is negative. Hence, the parabola opens towards the left.
On comparing this equation with y² = –4ax, we obtain
–4a = –8
⇒ a = 2
∴ Coordinates of the focus = (–a, 0) = (–2, 0)
Since the given equation involves y², the axis of the parabola is the x-axis.
Equation of directrix, x = a i.e., x = 2
Length of latus rectum = 4a = 8

4. Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = – 16y
Solution
The given equation is x² = –16y.
Here, the coefficient of y is negative. Hence, the parabola opens downwards.
On comparing this equation with x² = – 4ay, we obtain
–4a = –16
⇒ a = 4
∴ Coordinates of the focus = (0, –a) = (0, –4)
Since the given equation involves x², the axis of the parabola is the y-axis.
Equation of directrix, y = a i.e., y = 4
Length of latus rectum = 4a = 16

5. Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y² = 10x
Solution
The given equation is y² = 10x.
Here, the coefficient of x is positive. Hence, the parabola opens towards the right.
On comparing this equation with y² = 4ax, we obtain 
4a = 10
⇒ a = 5/2 
∴ Coordinates of the focus = (a, 0) = (5/2, 0) 
Since the given equation involves y² , the axis of the parabola is the x - axis.  
Equation of directrix, x = -a, i.e., x = -5/2 
Length of latus rectum  = 4a = 10

6. Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = –9y .
Solution
The given equation is x² = –9y.
Here, the coefficient of y is negative. Hence, the parabola opens downwards.
On comparing this equation with x² = –4ay, we obtain 
-4a = -9
⇒ b = 9/4 
∴ Coordinates of the focus = (0, -a) = (0, -9/4) 
Since the given equation involves x² , the axis of the parabola is the y - axis. 
Equation of directrix, y = a, i.e., y = 9/4 
Length of latus rectum = 4a = 9

7. Find the equation of the parabola that satisfies the following conditions: Focus (6, 0); directrix x = –6
Solution
Focus (6, 0); directrix, x = –6
Since the focus lies on the x-axis, the x-axis is the axis of the parabola.
Therefore, the equation of the parabola is either of the form y² = 4ax or
y² = – 4ax.
It is also seen that the directrix, x = –6 is to the left of the y-axis, while the focus (6, 0) is to the right of the y-axis. Hence, the parabola is of the form y² = 4ax.
Here, a = 6
Thus, the equation of the parabola is y² = 24x.

8. Find the equation of the parabola that satisfies the following conditions: Focus (0, –3); directrix y = 3
Solution
Focus = (0, –3); directrix y = 3
Since the focus lies on the y-axis, the y-axis is the axis of the parabola.
Therefore, the equation of the parabola is either of the form x² = 4ay or
x² = – 4ay.
It is also seen that the directrix, y = 3 is above the x-axis, while the focus
(0, –3) is below the x-axis. Hence, the parabola is of the form x² = –4ay.
Here, a = 3
Thus, the equation of the parabola is x² = –12y.

9. Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0); focus (3, 0)
Solution
Vertex (0, 0); focus (3, 0)
Since the vertex of the parabola is (0, 0) and the focus lies on the positive x-axis, x-axis is the axis of the parabola, while the equation of the parabola is of the form y² = 4ax.
Since the focus is (3, 0), a = 3.
Thus, the equation of the parabola is y² = 4×3×x, i.e., y² = 12x

10. Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) focus (–2, 0)
Solution
Vertex (0, 0) focus (–2, 0)
Since the vertex of the parabola is (0, 0) and the focus lies on the negative x-axis, x-axis is the axis of the parabola, while the equation of the parabola is of the form y² = –4ax.
Since the focus is (–2, 0), a = 2.
Thus, the equation of the parabola is y² = –4(2)x, i.e., y² = –8x

11. Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) passing through (2, 3) and axis is along x-axis
Solution
Since the vertex is (0, 0) and the axis of the parabola is the x-axis, the equation of the parabola is either of the form y² = 4ax or y² = –4ax.
The parabola passes through point (2, 3), which lies in the first quadrant.
Therefore, the equation of the parabola is of the form y² = 4ax, while point
(2, 3) must satisfy the equation y² = 4ax. 
∴ 3² = 4a(2) ⇒ a = 9/8 
Thus, the equation of the parabola is  

12. Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis
Solution
Since the vertex is (0, 0) and the parabola is symmetric about the y-axis, the equation of the parabola is either of the form x² = 4ay or x² = –4ay.
The parabola passes through point (5, 2), which lies in the first quadrant.
Therefore, the equation of the parabola is of the form x² = 4ay, while point
(5, 2) must satisfy the equation x² = 4ay. 
∴ (5)² = 4×a ×2 ⇒ 25 = 8a ⇒ a = 25/8 
Thus, the equation of the parabola is  

Exercise 11.3

1. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse x²/36 + y²/16 = 1 
Solution
The given equation is x²/36 + y²/16 = 1.
Here, the denominator of x²/36 is greater than the denominator of y²/16 . 
Therefore, the major axis is along the x - axis, while the minor axis is along the y - axis. 
On comparing the given equation with x²/a² + y²/b²  = 1 , we obtain a = 6 and b = 4. 

2. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse x²/4 + y²/25 = 1 
Solution

3. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse x²/16 + y²/9 = 1.
Solution

4. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse x²/25 + y²/100 = 1 
Solution
The given equation is

Here, the denominator of y²/100 is greater than the denominator of x²/25. 
Therefore, the major axis is along the y - axis, while the minor axis is along the x-axis.
On comparing the given equation with x² /b²  + y² /a²  = 1, we obtain b = 5 and a = 10. 

5. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse x²/49 + y²/36 = 1 
Solution
The given equation is

Here, the denominator of x²/49 is greater than the denominator of y²/36. 
Therefore, the major axis is along the x - axis, while the minor axis is along the y - axis.  
On comparing the given equation with x²/a² + y²/b²  = 1, we obtain a = 7 and b = 6. 

6. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse x²/100 + y²/400 = 1
Solution
The given equation is

Here, the denominator of y² /400 is greater than the denominator of x²/100. 
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis. 
On comparing the given equation with x²/b² + y²/a² = 1, we obtain b = 10 and a = 20. 

7. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 36x² + 4y² = 144
Solution
The given equation is 36x² + 4y² = 144.
It can be written as 

Here, the denominator of y²/6² is greater than the denominator of x²/2².
Therefore, the major axis is along the y - axis, while the minor axis is along the x - axis. 
On comparing equation (1) with x²/b² + y²/a² = 1, we obtain b = 2 and a = 6. 

8. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 16x² + y² = 16
Solution
The given equation is 16x² + y² = 16.
It can be written as 

Here, the denominator of y²/4² is greater than the denominator of x²/1².
Therefore, the major axis is along the y - axis, while the minor axis is along the x - axis. 
On comparing equation (1) with x²/b² + y²/a² = 1, we obtain b = 1 and a = 4. 

Therefore,  
The coordinates of the foci are (0, ± √15). 
The coordinates of the vertices are (0, ±4). 
length of major axis = 2a = 8 
Length of minor axis = 2b = 2
Eccentricity, e = c/a = √15/4 
Length of latus rectum - 2b²/a = (2×1)/4 = 1/2

9. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4x² + 9y² = 36
Solution
The given equation is 4x² + 9y² = 36.
It can be written as

Therefore, 
The coordinates of the foci are (±√5, 0). 
The coordinates of the vertices are (±3, 0). 
Length of major axis = 2a = 6 
Length of minor axis = 2b = 4 
Eccentricity, e = c/a = √5/3 
Length of latus rectum = 2b²/a = (2× 4)/3 = 8/3

10. Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0).
Solution
Vertices (±5, 0), foci (±4, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the ellipse will be of the form x²/a² + y²/b² = 1 , where a is the semi-major axis.
Accordingly, a = 5 and c = 4.
It is known that a² = b² + c².
∴ 5² = b² + 4²
⇒ 25 = b² + 16
⇒ b² = 25 - 16
⇒ b = √9 = 3
Thus, the equation of the ellipse 

11. Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)
Solution
Vertices (0, ±13), foci (0, ±5)
Here, the vertices are on the y-axis.
Therefore, the equation of the ellipse will be of the form x² /a² + y² /b² = 1, where a is the semi-major axis.
Accordingly, a = 13 and c = 5.
It is known that  a² = b² + c².
∴ 13² = b² + 5²
⇒ 169 = b² + 25
⇒ b² = 169 - 25
⇒ b = √144 = 12
Thus, the equation of the ellipse is 

12. Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0)
Solution
Vertices (±6, 0), foci (±4, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the ellipse will be of the form x² /a² + y² /b² = 1, where a is the semi-major axis.
Accordingly, a = 6, c = 4.
It is known that a² = b² + c².
∴ 6² = b² + 4²
⇒ 36 = b² + 16
⇒ b² = 36 - 16
⇒ b = √ 20
Thus, the equation of the ellipse is 

13. Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3, 0), ends of minor axis (0, ±2)
Solution
Ends of major axis (±3, 0), ends of minor axis (0, ±2)
Here, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form x²/a² + y²/b² = 1, where a is the semi-major axis.
Accordingly, a = 3 and b = 2.
Thus, the equation of the ellipse is 

14. Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0, ± √5), ends of minor axis (±1, 0)
Solution
Ends of major axis (0, ± √5), ends of minor axis (±1, 0)
Here, the major axis is along the y-axis.
Therefore, the equation of the ellipse will be of the form x² /a² + y² /b² = 1, where a is the semi-major axis.
Accordingly, a = √5 and b = 1.
Thus, the equation of the ellipse is  

15. Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (± 5, 0)
Solution
Length of major axis = 26; foci = (±5, 0).
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form  x² /a² + y² /b² = 1, where a is the semi-major axis.
Accordingly,
2a = 26
⇒ a = 13 and c = 5.
It is known that a² = b² + c².
∴ 13² = b² + 5² 
⇒ 169 = b² + 25
⇒ b² = 169 - 25
⇒ b = √144 = 12
Thus, the equation of the ellipse is 

16. Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)
Solution
Length of minor axis = 16; foci = (0, ± 6).
Since the foci are on the y-axis, the major axis is along the y-axis.
Therefore, the equation of the ellipse will be of the form x² /a² + y² /b² = 1, where a is the semi-major axis.
Accordingly,
2b = 16
⇒ b = 8 and c = 6.
It is known that a² = b² + c².
∴ a² = 8² + 6² = 64 + 36 = 100 
⇒ a = √100 = 10
Thus, the equation of the ellipse is

17. Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), a = 4
Solution
Foci (± 3, 0), a = 4
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form x²/a² + y²/b² = 1, where a is the semi-major axis.
Accordingly, c = 3 and a = 4.
It is known that a² = b² + c².
∴ 4² = b² + 3² 
⇒ 16 = b² + 9 
⇒ b² = 16 - 9 = 7
Thus, the equation of the ellipse is x²/16 + y²/7 = 1.

18. Find the equation for the ellipse that satisfies the given conditions: b = 3, c = 4, centre at the origin; foci on the x axis.
Solution
It is given that b = 3, c = 4, centre at the origin; foci on the x axis.
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form x²/a² + y²/b² = 1, where a is the semi-major axis.
Accordingly, b = 3, c = 4.
It is known that a² = b² + c².
∴ a² = 3² + 4² = 9 + 16 = 25 
⇒ a = 5 
Thus, the equation of the ellipse is

19. Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
Solution
Since the centre is at (0, 0) and the major axis is on the y-axis, the equation of the ellipse will be of the form

On solving equations (2) and (3), we obtain b² = 10 and a² = 40. 
Thus, the equation of the ellipse is x²/10 + y²/40 = 1 or 4x² + y² = 40.

20. Find the equation for the ellipse that satisfies the given conditions: Major axis on the x-axis and passes through the points (4, 3) and (6, 2).
Solution
Since the major axis is on the x - axis, the equation of the ellipse will be of the form 

On solving equations (2) and (3), we obtain a² = 52 and b² = 13.
Thus, the equation of the ellipse is x²/52 + y²/13 = 1 or x² + 4y² = 52.

Exercise 11.4

1. Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola x²/16 - y²/9 = 1 
Solution
The given equation is

On comparing the equation with the standard equation of hyperbola i.e., x² /a² - y² /b² = 1, we obtain a = 4 and b = 3.
We know that a² + b² = c².
∴ c² = 4² + 3² = 25
⇒ c = 5
Therefore, 
The coordinates of the foci are (± 5, 0). 
The coordinates of the vertices are (± 4, 0). 
Eccentricity, e = c/a = 5/4 
Length of latus rectum = 2b²/a = (2×9)/4 = 9/2.

2. Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola y²/9 - x²/27 = 1 
Solution
The given equation is

 

On comparing this equation with the standard equation of hyperbola i.e., y²/a² - x²/b² = 1, we obtain a = 3 and b = √27.
We know that a² + b² = c².
∴ c² = 3² + (√27)² = 9 + 27 = 36
⇒ c = 6
Therefore, 
The coordinates of the foci are (0, ± 6).
The coordinates of the vertices are (0, ± 3).
Eccentricity, e = c/a = 6/3 = 2
Length of latus rectum = 2b²/a = (2× 27)/3 = 18

3. Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 9y² – 4x² = 36.
Solution
The given equation is 9y² – 4x² = 36.
It can be written as
9y² – 4x² = 36 

On comparing equation (1) with the standard equation of hyperbola i.e., y²/a² - x²/b² = 1, we obtain a = 2 and b = 3.
We know that a² + b² = c².
∴ c² = 4 + 9 = 13
⇒ c = √13
Therefore, 
The coordinates of the foci are (0, ± √13)
The coordinates of the vertices are (0, ± 2).
Eccentricity,  e = c/a = √13/2
Length of latus rectum  = 2b²/a = (2× 9)/2 = 9

4. Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 16x² – 9y² = 576 
Solution
The given equation is 16x² – 9y² = 576.
It can be written as

On comparing equation (1) with the standard equation of hyperbola i.e., x²/a² - y²/b² = 1 , we obtain a = 6 and b = 8.
We know that a² + b² = c².
∴ c² = 36 + 64 = 100 
⇒ c = 10
Therefore, 
The coordinates of the foci are (± 10, 0). 
The coordinates of the vertices are ( ±6, 0). 
Eccentricity, e = c/a = 10/6 = 5/3
Length of latus rectum =2b²/a = (2 × 64)/6 = 64/3

5. Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y² – 9x² = 36
Solution
The given equation is 5y² - 9x² = 36. 

6. Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 49y² – 16x² = 784
Solution
The given equation is 49y² – 16x² = 784.
It can be written as
49y² – 16x² = 784 

On comparing equation (1) with the standard equation of hyperbola i.e., y² /a² - x² /b² = 1, we obtain a = 4 and b = 7 
We know that a² + b² = c².
∴ c² = 16 + 49 = 65 
⇒ c = √65 
Therefore, 
The coordinates of the foci are (0, ± √65). 
The coordinates of the vertices are (0, ± 4). 
Eccentricity, e = c/a = √65/4 
Length of latus rectum  = 2b²/a = (2× 49)/4 = 49/2

7.  Find the equation of the hyperbola satisfying the give conditions: Vertices (±2, 0), foci (±3, 0).
Solution
Vertices (±2, 0), foci (±3, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form x²/a² - y²/b²  = 1. 
Since the vertices are (±2, 0), a = 2.
Since the foci are (±3, 0), c = 3.
We know that a² + b² = c².
∴ 2² + b² = 3² 
b² = 9 - 4 = 5 
Thus, the equation of the hyperbola is x²/4 - y² /5 = 1.

8. Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±5), foci (0, ± 8).
Solution
Vertices (0, ±5), foci (0, ±8)
Here, the vertices are on the y-axis.
Therefore, the equation of the hyperbola is of the form y²/a² - x²/b²  = 1.
Since the vertices are (0, ±5), a = 5.
Since the foci are (0, ±8), c = 8.
We know that a² + b² = c². 
∴ 5² + b² = 8² 
b² = 64 - 25 = 39 
Thus, the equation of the hyperbola is y²/25 - x²/39 = 1.

9. Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±3), foci (0, ± 5) . 
Solution
Vertices (0, ±3), foci (0, ±5)
Here, the vertices are on the y-axis.
Therefore, the equation of the hyperbola is of the form y²/a² - x² /b²  = 1.
Since the vertices are (0, ±3), a = 3.
Since the foci are (0, ±5), c = 5.
We know that a² + b² = c².
∴3² + b² = 5²
⇒ b² = 25 – 9 = 16
Thus, the equation of the hyperbola is y²/9 - x²/16 = 1.

10. Find the equation of the hyperbola satisfying the give conditions: Foci (±5, 0), the transverse axis is of length 8.
Solution
Foci (±5, 0), the transverse axis is of length 8.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form x²/a² - y²/b²  = 1.
Since the foci are (±5, 0), c = 5.
Since the length of the transverse axis is 8,
2a = 8
⇒ a = 4.
We know that a² + b² = c².
∴ 4² + b² = 5²
⇒ b² = 25 – 16 = 9
Thus, the equation of the hyperbola is x²/16 - y²/9 = 1.

11. Find the equation of the hyperbola satisfying the give conditions: Foci (0, ±13), the conjugate axis is of length 24.
Solution
Foci (0, ±13), the conjugate axis is of length 24.
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form y²/a² - x²/b² = 1.
Since the foci are (0, ±13), c = 13.
Since the length of the conjugate axis is 24,
2b = 24
⇒ b = 12.
We know that a² + b² = c².
∴ a² + 12² = 13²
⇒ a² = 169 – 144 = 25
Thus, the equation of the hyperbola is y²/25  - x²/144 = 1.

12. Find the equation of the hyperbola satisfying the give conditions: Foci (± 3√5, 0), the latus rectum is of length 8.
Solution
Foci (± 3√5, 0), the latus rectum is of length 8. 
Here, the foci are on the x - axis. 
Therefore, the equation of the hyperbola is of the form x²/a² - y²/b² = 1. 
Since the foci are (±3√5, 0), c = ± 3√5 . 
Length of latus rectum  = 8 
⇒ 2b² /a = 8 
⇒ b² = 4a 
We know that a² + b² = c² . 
∴ a² + 4a = 45
⇒ a² + 4a – 45 = 0
⇒ a² + 9a – 5a – 45 = 0
⇒ (a + 9) (a – 5) = 0
⇒⇒ a = –9, 5
Since a is non-negative, a = 5.
∴ b² = 4a = 4 × 5 = 20
Thus, the equation of the hyperbola is x²/25 - y²/20 = 1.

13. Find the equation of the hyperbola satisfying the give conditions: Foci (±4, 0), the latus rectum is of length 12 . 
Solution
Foci (±4, 0), the latus rectum is of length 12.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form x²/a² - y²/b² = 1 .
Since the foci are (±4, 0), c = 4.
Length of latus rectum = 12  
⇒ 2b²/a = 12 
⇒ b² = 6a
We know that a² + b² = c².
∴ a² + 6a = 16
⇒ a² + 6a – 16 = 0
⇒ a² + 8a – 2a – 16 = 0
⇒ (a + 8) (a – 2) = 0
⇒ a = –8, 2
Since a is non-negative, a = 2.
∴ b² = 6a = 6×2 = 12
Thus, the equation of the hyperbola is x²/4 - y²/12 = 1.

14. Find the equation of the hyperbola satisfying the give conditions: Vertices (±7, 0), e = 4/3
Solution
Vertices (±7, 0), e = 4/3
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form x²/a² - y²/b² = 1.
Since the vertices are (±7, 0), a = 7.
It is given that e = 4/3 

15. Find the equation of the hyperbola satisfying the give conditions: Foci (0, ±√10), passing through (2, 3) .
Solution
Foci (0, ±√10), passing through (2, 3)
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form y²/a² - x²/b² = 1.
Since the foci are (0, ± √10), c = √10 .
We know that a² + b² = c².
∴ a² + b² = 10
⇒ b² = 10 – a² … (1)
Since the hyperbola passes through point (2, 3), 
9/a² - 4/b² = 1  ...(2) 
From equations (1) and (2), we obtain  

⇒ 9(10 - a² ) - 4a² = a² (10 - a²)
⇒ 90 - 9a² - 4a² = 10a² - a⁴ 
⇒ a⁴ - 23a² + 90 = 0 
⇒ a⁴ - 18a² - 5a² + 90 = 0 
⇒ a² (a² - 18) - 5(a² - 18) = 0
⇒ (a² - 18)(a² - 5) = 0 
⇒ a² = 18 or 5 
In hyperbola, c > a, i.e., c² > a² 
a² = 5 
b² = 10 - a² = 10 - 5 = 5 
Thus, the equation of the hyperbola is y²/5 - x²/5 = 1.

Miscellaneous Solutions

1. If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
Solution
​​​​​​​The origin of the coordinate plane is taken at the vertex of the parabolic reflector in such a way that the axis of the reflector is along the positive x-axis.
This can be diagrammatically represented as 

The equation of the parabola is of the form y² = 4ax (as it is opening to the right).
Since the parabola passes through point A (10, 5), 10² = 4a(5)
⇒ 100 = 20a 
⇒ a = 100/20 = 5
Therefore, the focus of the parabola is (a, 0) = (5, 0), which is the mid-point of the diameter.
Hence, the focus of the reflector is at the mid-point of the diameter.

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
Solution
The origin of the coordinate plane is taken at the vertex of the arch in such a way that its vertical axis is along the negative y-axis.
This can be diagrammatically represented as 

The equation of the parabola is of the form x² = −4ay (as it is opening downwards).
It can be clearly seen that the parabola passes through point (5/2,−10). 
(5/2)² = 4a (10) 
⇒ 4a = 25/(4 × 10) = 5/8
Therefore, the arch is in the form of a parabola whose equation is x² = (-5/8)y . 
When y = -2, x² = (-5/8) (-2) 
⇒ x² = 5/4 
⇒ x = √5/2 
∴ AB = 2× (√5/2) m = √5 m = 2.23 m (approx.) 
Hence , when the arch is  2 m from the vertex of the parabola, its width is approximately 2.23m.

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
Solution
The vertex is at the lowest point of the cable. The origin of the coordinate plane is taken as the vertex of the parabola, while its vertical axis is taken along the positive y-axis. This can be diagrammatically represented as

Here, AB and OC are the longest and the shortest wires, respectively, attached to the cable.
DF is the supporting wire attached to the roadway, 18 m from the middle.
Here, AB = 30 m, OC = 6 m, and BC = 100/2 = 50 m.
The equation of the parabola is of the form x² = 4ay (as it is opening upwards).
The coordinates of point A are (50, 30 – 6) = (50, 24).
Since A (50, 24) is a point on the parabola, 

⇒ y = 3.11 (approx) 
DE = 3.11 m 
DF = DE + EF = 3.11m + 6m = 9.11 m 
Thus, the length of the supporting wire attached to the roadway 18 m from the middle is approximately 9.11 m.

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
Solution
Since the height and width of the arc from the centre is 2 m and 8 m respectively, it is clear that the length of the major axis is 8 m, while the length of the semi-minor axis is 2 m.
The origin of the coordinate plane is taken as the centre of the ellipse, while the major axis is taken along the x-axis. Hence, the semi-ellipse can be diagrammatically represented as

The equation of the semi - ellipse will be of the form x²/a² + y²/b² = 1, y ≥ 0 , where a is the semi- major axis  
Accordingly, 2a = 8
⇒ a = 4
b = 2
Therefore, the equation of the semi-ellipse is x²/16 + y² /4 = 1, y ≥ 0 ...(1)
Let A be a point on the major axis such that AB = 1.5 m.
Draw AC⊥ OB.
OA = (4 – 1.5) m = 2.5 m
The x-coordinate of point C is 2.5.
On substituting the value of x with 2.5 in equation (1), we obtain

⇒ y² = 2.4375 
⇒ y = 1.56 (approx.) 
DAC = 1.56 m 
Thus, the height of the arch at a point 1.5 m from one end is approximately 1.56 m.

5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.
Solution
Let AB be the rod making an angle θ with OX and P (x, y) be the point on it such that AP = 3 cm.
Then, PB = AB – AP = (12 – 3) cm = 9 cm [AB = 12 cm]
From P, draw PQ ⊥OY and PR ⊥ OX. 

6. Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of its latus rectum.
Solution
The given parabola is x² = 12y.
On comparing this equation with x² = 4ay, we obtain 4a = 12 ⇒ a = 3
∴The coordinates of foci are S (0, a) = S (0, 3)
Let AB be the latus rectum of the given parabola.
The given parabola can be roughly drawn as 

Thus, the required area of the triangle is 18 unit².

7. A man running a racecourse notes that the sum of the distances from the two flag posts form him is always 10 m and the distance between the flag posts is 8 m. find the equation of the posts traced by the man.
Solution
Let A and B be the positions of the two flag posts and P(x, y) be the position of the man. Accordingly, PA + PB = 10.
We know that if a point moves in a plane in such a way that the sum of its distances from two fixed points is constant, then the path is an ellipse and this constant value is equal to the length of the major axis of the ellipse.
Therefore, the path described by the man is an ellipse where the length of the major axis is 10 m, while points A and B are the foci.
Taking the origin of the coordinate plane as the centre of the ellipse, while taking the major axis along the x-axis, the ellipse can be diagrammatically represented as

The equation of the ellipse will be of the form x²/a² + y²/b² = 1, where a is the semi - major axis 
Accordingly, 2a = 10
a = 5
Distance between the foci (2c) = 8
c = 4
On using the relation c = √(a² - b²) , we obtain
4 = √(25 - b²)
⇒ 16 = 25 - b²
⇒ b² = 25 - 16 = 9
⇒ b = 3 
Thus, the equation of the path traced by the man is x²/25 + y²/9 = 1.

8. An equilateral triangle is inscribed in the parabola y² = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
Solution
Let OAB be the equilateral triangle inscribed in parabola y² = 4ax.
Let AB intersect the x-axis at point C. 

 

Let OC = k 
From the equation of the given parabola, we have y² = 4ak ⇒ y = ± 2√ak 
∴ The respective coordinates of points A and B are (k, 2√ak), and (k, -2√ak) 
AB = CA + CB = 2√ak + 2√ak = 4√ak 
Since OAB is an equilateral triangle, OA² = AB² . 
∴ k² + (2√ak)² = (4√ak)² 
⇒ k² + 4ak = 16ak 
⇒ k² = 12ak 
⇒ k = 12a 
∴ AB = 4√ak = 4√a × 12a = 4√12a² = 8√3a 
Thus, the side of the equilateral triangle inscribed in parabola y² = 4 ax is 8√3a.