BITSAT Mathematics Conic Sections MCQs

Question: Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1, 2) and eccentricity √3

• a) 7x² – 2y² + 12xy – 2x – 14 – 22 = 0
• b) 7x² + 2y² – 12xy – 2x + 14 – 22 = 0
• c) 7x² + 2y² + 12xy – 2x – 14 – 22 = 0
• d) 7x² – 2y² + 12xy – 2x + 14 – 22 = 0

Answer: 7x² – 2y² + 12xy – 2x + 14 – 22 = 0

Question: Find the parametric coordinates of any point of the circle x² + y² + 2x – 3y – 4 = 0

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: If the tangent at the point P(x₁, y₁) to the parabola y² = 4ax meets the parabola y² = 4a(x + b) at Q and R, then the mid-point of QR is

• a) (x₁ + b, y₁ + b)
• b) (x₁ – b, y₁ – b)
• c) (x₁, y₁)
• d) (x₁ + b, y₁)

Answer: (x₁, y₁)

Question: If the line 3x + ay – 20 = 0 cuts the circle x² + y² = 25 at real distinct or coincident points, then a belongs to the interval

• a) [- √7, √7]
• b) (- √7, 1/7)
• c)

  

• d) None of these

Answer:

Question: The total number of common tangents to the two circles x² + y² – 2x – 6y + 9 = 0 and x² + y² + 6x – 2y + 1 = 0, is -

• a) 1
• b) 2
• c) 3
• d) 4

Answer: 4

Question: The equation of the tangent to the elipse x² + 4y² = 25 at the point whose ordinate is 2 is

• a) x + 2y = 5 or 2x - y = 5
• b) 3x +8y = 25 or 8y -3x = 25
• c) 3x + 2y =15 or 3y - 2x =15
• d) None of these

Answer: 3x +8y = 25 or 8y -3x = 25

Question: An equilateral triangle is inscribed in the circle x² + y² = a² with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex ?

• a) 2x – a = 0
• b) x + a = 0
• c) 2x + a = 0
• d) 3x – 2a = 0

Answer: 2x + a = 0

Question: The equation of one of the common tangents to the parabola y² = 8x and x² + y² -12x + 4 = 0 is

• a) y = –x + 2
• b) y = x – 2
• c) y = x + 2
• d) None of these

Answer: y = x + 2

Question: The line joining (5, 0) to ( (10cos θ, 10sin θ) is divided internally in the ratio 2 : 3 at P. If θ varies, then the locus of P is

• a) A pair of straight
• b) A circle lines
• c) A straight line
• d) None of these

Answer: A circle lines

Question: The number of integral values of λ for which x² + y² + λx + (1- λ)y + 5 = 0 is the equation of a circle whose radius cannot exceed 5, is

• a) 14
• b) 18
• c) 16
• d) None of these

Answer: 16

Question: An ellipse has OB as semi minor axis, F and F ' its focii and the angle FBF ' is a right angle. Then the eccentricity of the ellipse is

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: If the line 2x – 3y = k touches the parabola y² = 6x, then find the value of k

• a) –15/4
• b) –27/4
• c) –1/4
• d) –3/4

Answer: –27/4

Question: S and T are the foci of an ellipse and B is an end of the minor axis. If STB is an equilateral triangle, then the eccentricity of the ellipse is

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: The length of the tangent drawn from any point on the circle x² + y² + 2fy + λ = 0 to the circle x²+ y² + 2fy + μ = 0, where μ > λ > 0, is

• a)

  

• b)

  

• c)

  

• d) μ + λ

Answer:

Question: Find the eccentricity of the conic represented by x² – y² – 4x + 4y + 16 = 0

• a) 2
• b) √2
• c) 2 √2
• d) 3 √2

Answer: √2

Question: The length of the latus-rectum of the parabola whose focus is

 and directrix is

  is

• a) 

  

• b)

  

• c)

  

• d)

  

Answer:

Question: The equation of the ellipse with focus at  and  as one directrix is

• a)

  

• b)

  

• c)

  

• d) None of these

Answer:

Question: For what value of k the circles x² + y² + 5x + 3y +7 = 0 and x² + y² – 8x + 6y + k = 0 cuts orthogonally

• a) 16
• b) –18
• c) – 13
• d) – 10

Answer: –18

Question: If the lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to a circle, then the radius of the circle is

• a) 3/2
• b) 3/4
• c) 1/10
• d) 1/20

Answer: 3/4

Question: If the line 2x – 1 = 0 is the directrix of the parabola y² – kx + 6 = 0 then one of the values of k is

• a) – 6
• b) 6
• c) 1/4
• d) – 1/4

Answer: – 6

Question: The line ax + by = 1 cuts ellipse cx² + dy² = 1 only once if

• a) ca ² + db² =1
• b) 

  

• c)

  

• d) ac² + bd² = 1

Answer:

Question: Find the equation of chord of the circle x² + y²= 8x bisected at the point (4, 3)

• a) y = 3
• b) y = 1
• c) y = 6
• d) y = 7

Answer: y = 3

Question: Find the vertex of the parabola x² – 8y – x + 19 = 0.

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: Which of the following lines, is a normal to the parabola y² = 16x ?

• a) y = x – 11 cosθ – 3 cos3θ
• b) y = x – 11 cosθ – cos3θ
• c) y = (x – 11) cosθ + cos3θ
• d) y = (x – 11) cosθ – cos3θ

Answer: y = (x – 11) cosθ – cos3θ

Question: For what value of λ does the line y = x + λ touches the ellipse 9x² + 16y² =144.

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: The length of the semi-latus rectum of an ellipse is one thrid of its major axis, its eccentricity would be

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: An equilateral triangle is inscribed in the circle x² + y² = a² with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex ?

• a) 2x – a = 0
• b) x + a = 0
• c) 2x + a = 0
• d) 3x – 2a = 0

Answer: 2x + a = 0

Question: The equation of one of the common tangents to the parabola y² = 8x and x² + y² -12x + 4 = 0 is

• a) y = –x + 2
• b) y = x – 2
• c) y = x + 2
• d) None of these

Answer: y = x + 2

Question: The line joining (5, 0) to ( (10cos q, 10sin q) is divided internally in the ratio 2 : 3 at P. If θ varies, then the locus of P is

• a) A pair of straight
• b) A circle lines
• c) A straight line
• d) None of these

Answer: A circle lines

Question: The number of integral values of λ for which x² + y² + λx + (1- λ)y + 5 = 0 is the equation of a circle whose radius cannot exceed 5, is

• a) 14
• b) 18
• c) 16
• d) None of these

Answer: 16

Question: The lengths of the tangent drawn from any point on the circle 15x² +15y² - 48x + 64y = 0 to the two circles 5x² + 5y² – 24x + 32y + 75 = 0 and 5x² + 5y² – 48x + 64y + 300 = 0 are in the ratio of

• a) 1 : 2
• b) 18
• c) 16
• d) None of these

Answer: 1 : 2

Question: The length of the chord x + y = 3 intercepted by the circle x² + y² - 2x - 2y - 2 = 0 is

• a)

  

• b)

  

• c)

  √14

• d)

  

Answer:

√14

Question: The locus of the point of intersection of two tangents to the parabola y² = 4ax, which are at right angle to one another is

• a) x² + y² = a²
• b) ay² =x
• c) x + a = 0
• d)

  

Answer: x + a = 0

Question: The parabola having its focus at (3, 2) and directrix along the y-axis has its vertex at

• a) (2, 2)
• b)

  

• c)

  

• d)

  

Answer:

Question: Let S be the focus of the parabola y² = 8x and PQ be the common chord of the circle x² + y² – 2x – 4y = 0 and the given parabola. The area of ΔPQS is

• a) 4 sq units
• b) 3 sq units
• c) 2 sq units
• d) 8 sq units

Answer: 4 sq units

Question: The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x = 4 , then the equation of the ellipse is:[

• a) 4x² + 3y² = 1
• b) 3x² + 4y² = 12
• c) 4x² + 3y² = 12
• d) 3x² + 4y² = 1

Answer: 3x² + 4y² = 12

Question: Area of the circle in which a chord of length √2 makes an angle p/2 at the centre, is

• a) π/2 sq units
• b) 2π sq units
• c) π sq units
• d) π/4 sq units

Answer: π sq units

Question: The angle of intersection of the two circles x² + y² – 2x – 2y = 0 and x² + y² = 4, is

• a) 30º
• b) 60º
• c) 90º
• d) 45º

Answer: 45º

Question: An arch of a bridge is semi-elliptical with major axis horizontal. If the length the base is 9 meter and the highest part of the bridge is 3 meter from the horizontal; the best approximation of the height of the arch. 2 meter from the centre of the base is

• a) 11/4 m
• b) 8/3 m
• c) 7/2 m
• d) 2 m

Answer: 8/3 m

Question: A pair of tangents are drawn from the origin to the circle x² + y²+ 20 (x + y) + 20 = 0, then the equation of the pair of tangent are

• a) x² + y² - 5xy = 0
• b) x² + y² + 2x + y = 0
• c) x² + y² – xy + 7 = 0
• d) 2x² + 2y² + 5xy = 0

Answer: 2x² + 2y² + 5xy = 0