JEE Mathematics Circles MCQs Set C

Question: Equation of the circle passing through the origin and through the points of intersection of the circle

x² + y²– 2x + 4y – 20 = 0 and the line x + y – 1 = 0 is

• a) x² + y² – 22x –16y = 0
• b) x² + y² - 20x +15y = 0
• c) x² + y² + 33x + 33y = 0
• d) None of these

Answer: x² + y² – 22x –16y = 0

Question: Equation of the circle concentric with the circle x² + y² – 3x+ 4y – c = 0 and passing through the point (–1, – 2).

• a) x² + y² - 3x + 4y = 0
• b) x² + y² – 7x + 7y = 0
• c) x² + y² - 3x - 4y = 0
• d) x² + y² + 3x + 4y = 0

Answer: x² + y² - 3x + 4y = 0

Question: The straight line (x – 2) + (y + 3) = 0 cuts the circle (x – 2)² + ( y – 3)² = 11 at

• a) No points
• b) Two points
• c) One point
• d) None of these

Answer: No points

Question: If the centre of a circle which passing through the points of intersection of the circle x² + y² – 6x + 2y + 4 = 0 and x² + y² + 2x – 4y – 6 = 0 is on the line y = x, then the equation of the circle is

• a) 7x² + 7y² -10x -10y -12 = 0
• b) x² + y² + 2x + 4y = 0
• c) 5x² + 5y² -10x -10y = 0
• d) 2x² + 2y² -3x -3y = 0

Answer: 7x² + 7y² -10x -10y -12 = 0

Question: Centre of the circle whose radius is 3 and which touches the circle x² + y² – 4x – 6y – 12 = 0 internally at the point (–1, –1) is

• a) (4/7, 7/5)
• b) (5/7, 7/5)
• c) (5/7, 2/5)
• d) none of these

Answer: (4/7, 7/5)

Question: The lines 2x – 3y = 5 and 3x – 4y = 7 are diameters of a circle of area 154 sq.units.Then the equation of the circle is

• a) x² + y² - 2x + 2y = 47 .
• b) x² + y² + 2x - 2y = 62
• c) x² + y² - 2x + 2y = 62
• d) x² + y² + 2x - 2y = 47

Answer: x² + y² - 2x + 2y = 47 .

Question: If the lines 2x + 3y +1 = 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference 10p, then the equation of the circle is

• a) x² + y² - 2x + 2y - 23 = 0
• b) x² + y² + 2x + 2y - 23 = 0
• c) x² + y² + 2x - 2 y - 23 = 0
• d) x² + y² - 2x - 2y - 23 = 0

Answer: x² + y² - 2x + 2y - 23 = 0

Question: The point diametrically opposite to the point P(1, 0) on the circle x² + y² + 2x + 4y – 3 = 0 is

• a) (–3, –4)
• b) (3, – 4)
• c) (–3, 4)
• d) (3, 4)

Answer: (–3, –4)

Question: The area of a circle centred at (1, 2) and passing through (4, 6) is (in sq. units) is

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question: A circle of radius 5 touches another circle x² + y² –2x – 4y –20 = 0 at (5, 5) then its equation is

• a) x² + y² – 18x – 16y + 120 = 0
• b) x² + y² – 18x + 16y + 120 = 0
• c) x² + y² + 18x + 16y + 120 = 0
• d) None of these

Answer: x² + y² – 18x – 16y + 120 = 0

Question: The circle x² + y² – 8x + 4y + 4 = 0 touches :

• a) y-axis only
• b) None of these
• c) x-axis only
• d) both

Answer: y-axis only

Question: The line 3x – 2y = k meets the circle x² + y² = 4r² at only one point then the value k² is

• a) 52r²
• b) 32r²
• c) 50 r²
• d) none of these

Answer: 52r²

More Questions..................................

Question: If (– 3, 2) lies on the circle x² + y² + 2gx + 2fy + c = 0, which is concentric with the circle x² + y² + 6x + 8y – 5 = 0, then c is equal to

• a) – 11
• b) 100
• c) 11
• d) 24

Answer: – 11

Question:

• a) 3/4
• b) 4/3
• c) 1
• d) 12

Answer: 3/4

Question: The length of intercept, the circle

x² + y² + 10x – 6y + 9 = 0 makes on the x-axis is :

• a) 8
• b) 4
• c) 2
• d) 6

Answer: 8

Question: If (x_i, 1/x_i =1, 2, 3, 4 are four distinct points on a circle, then the value of x1 . x2 . x3 . x4 is :

• a) 1
• b) 4
• c) – 1
• d) 0

Answer: 1

Question: If the line x + 2by + 7 = 0 is a diameter of the circle x² + y² – 6x + 2y =0, then b =

• a) 5
• b) –5
• c) 3
• d) –1

Answer: 5

Question: If the circle x² + y² + 2gx + 2fy + c = 0 touches x-axis, then

• a) g² = c
• b) g² + f² = c
• c) g = f
• d) f² = c

Answer: g² = c

Question: The circle x² + y² + 4x – 4y + 4 = 0 touches

• a) x-axis and y-axis
• b) x-axis
• c) y-axis
• d) None of these

Answer: x-axis and y-axis

Question:

• a) (2, – 1)
• b) (– 1, 2)
• c) (3, 3)
• d) (– 2, 1)

Answer: (2, – 1)

Question: If a circle S(x, y) = 0 touches the line x + y = 5 at the point (2, 3) and S (1, 2)= 0, then radius of such circle is

• a) 1/√2
• b) 4 units
• c) 2 units
• d) 1/2 units

Answer: 1/√2

Question: In the figure OABC is a square of side 8 cm, then the equation of the smallest circle is

• a) None of these
• b) (x - 4)² + (y - 4)² =12
• c) (x - 4)² + (y - 4)² = 8
• d) (x - 4)² + (y - 4)² = 4

Answer: None of these

Question:

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

Question:

• a) 1 and 2 are correct
• b) 1 and 3 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 2 are correct

Question: If (a , 0) is a point on a diameter of circle x² + y² = 4, then x² – 4x – a² = 0 has

(1) exactly one real root in (– 1,0] greater than – 1
(2) exactly one real root in [2, 5] greater than –1
(3) two distinct roots greater than – 1
(4) two distinct root greater than 5

• a) 1, 2 and 3 are correct
• b) 2 and 4 are correct
• c) 1 and 2 are correct
• d) 1 and 3 are correct

Answer: 1, 2 and 3 are correct

Question: One diagonal of a square is the portion of x-axis intercepted by the circle x² + y² – 4x + 6y – 12 = 0. Then the y-coordinate of the extrimity above the x-axis of the other diagonal is

(1) ( 2, 4 )     (2) (2, – 4)
(3) (– 2, – 4) (4) (–2, 4)

• a) 1 and 2 are correct
• b) 1 and 3 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 2 are correct

Question: Statement-1 : Number of circles passing through (1, 4), (2, 3), (– 1, 6) is one. Statement-2 : Through 3 non collinear points in a plane only one circle can be drawn.

• a) Statement -1 is False, Statement-2 is True
• b) Statement -1 is True, Statement-2 is False.
• c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: Statement -1 is False, Statement-2 is True

Question:

Statement-1 : The circle x² + y² + 2ax + c = 0 , x² + y² + 2by + c = 0 touches each other if 1/a² + 1/b² = 1/c

Statement-2 : Two circles with centre C₁,C₂ and radii r₁, r₂ touch each other, if r₁ ± r₂ = C₁C₂

• a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
• b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• c) Statement -1 is False, Statement-2 is True.
• d) Statement -1 is True, Statement-2 is False.

Answer: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Question:  The equation of pair of tangents drawn from the point (0,1) to the circle x² + y² – 2x + 4y = 0 is –

• a) 4x² – 4y² + 6xy – 6x + 8y –4 = 0
• b) x² – y² + 3xy – 3x + 2y –1 = 0
• c) x² – y² + 6xy – 6x + 8y –4 = 0
• d) 4x² – 4y² + 6xy + 6x + 8y –4 = 0

Answer: 4x² – 4y² + 6xy – 6x + 8y –4 = 0

Question: Chord of contact with respect to point (2, 2) of circle x² + y² = 1 is -

• a) x + y = 1/2
• b) x + y + 1
• c) x – y = 1/2
• d) x + y = 2

Answer: x + y = 1/2

Question: Distance between the chord of contact with respect to point (0, 0) and (g, f) of circle x² + y² + 2gx + 2fy + c = 0 is

• a)

  

• b)

  

• c) 1/2 (g² + f² + c)
• d) g² + f² 

Answer:

Question: The locus of the centre of the circle which touches externally the circle x² + y² – 6x – 6y + 14 = 0 and also touches the y-axis, is -

• a) y² – 10x – 6y + 14 = 0
• b) x² – 6x – 10y + 14 = 0
• c) y² – 6x – 10y + 14 = 0
• d) x² – 10x – 6y + 14 = 0

Answer: y² – 10x – 6y + 14 = 0

Question: The two circles x² + y² = ax and x² + y² = c² (with c > 0) touch each other if -

• a) c = |a|
• b) 2a = |c|
• c) 2c = a
• d) None of these

Answer: c = |a|

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

• a) y = 3
• b) None
• c) 3y = 1
• d) 4x – 3y = 9

Answer: y = 3

Question: If lines y = x + 3 cuts the circle x² + y² = a2 in two points A and B, then equation of circle with AB as diameter is -

• a) x² + y² + 3x – 3y – a² + 9 = 0
• b) x² + y² + 3x – 3y + a² + 9 = 0
• c) x² + y² – 3x + 3y – a² + 9 = 0
• d) None of these

Answer: x² + y² + 3x – 3y – a² + 9 = 0

Question: The equation of the circle which passes through points of intersection of circle x² + y² + 4x – 5y + 3 = 0 and x² + y² + 2x + 3y – 3 = 0 and point (–3, 2) is -

• a)

  

• b)

  

• c) x² + y² – 13x + 8y + 3 = 0
• d) x² + y² + 8x + 13y – 3 = 0

Answer:

Question: If the circle x² + y² + 4x + 22y + c = 0 bisects the circumference of the circle x² + y² – 2x + 8y – d = 0, then c + d =

• a) 50
• b) 56
• c) 40
• d) 60

Answer: 50

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

• a) –18
• b) 4
• c) 18
• d) – 4

Answer: –18

Question: Equation of polar of point (4, 4) with respect to circle (x –1)² + (y – 2)² = 1 is

• a) 3x + 2y – 8 = 0
• b) 3x + 2y + 8 = 0
• c) 2x + 3y – 8 = 0
• d) 3x – 2y + 8 = 0

Answer: 3x + 2y – 8 = 0

Question: The pole of the line x/a + y/b = 1 with respect to circle x² + y² = c2 is

• a) (c²/a, c²b)
• b) (c²/a², c²b²)
• c) (c/a, c/b)
• d) None of these

Answer: (c²/a, c²b)

Question: The radical centre of the three circles

x² + y² = a²,(x – c)² + y² = a² and x² + (y – b)² = a² is -

• a) (c/2, b/2)
• b) (a/2, b/2)
• c) (b/2, c/2)
• d) None of these

Answer: (c/2, b/2)

Question: The equation of the radical axis of two circles

x² + y² – x + 1 = 0 and 3(x² + y²) + y – 1 = 0 is -

• a) 3x + y – 4 = 0
• b) 3x – y + 4 = 0
• c) 3x – y – 4 = 0
• d) None of these

Answer: 3x + y – 4 = 0

Question: The locus of the point, the chord of contact of tangents from which to the circle x² + y² = a² subtends a right angle at the centre is a circle of radius -

• a) √2a
• b) 2a
• c) a/2
• d) a²

Answer: √2a

Question: If a chord of the circle x² + y² = 8 makes equal intercepts of length a on the coordinate axes, then-

• a) | a | < 4
• b) | a | < 8
• c) | a | > 4
• d) None of these

Answer: | a | < 4

Question: The equation of a normal to the circle

x² + y² + 6x + 8y + 1 = 0 passing through (0, 0) is -

• a) 4x – 3y = 0
• b) 3x – 4y = 0
• c) 3x + 4y = 0
• d) 4x + 3y = 0

Answer: 4x – 3y = 0

Question: If the tangent to a circle x² + y² = 5 at point (1, –2) touches the circle x² +y² – 8x +6y+ 20 = 0, then its point of contact is-

• a) (3, –1)
• b) (5, 0)
• c) (–2, 1)
• d) (–1, –3)

Answer: (3, –1)

Question: Length of the tangent drawn from point (1, 5) to the circle

2x² + 2y² = 3 is -

• a) 7 √2 / 2
• b) 7
• c) 7 √2
• d) None of these

Answer: 7 √2 / 2

Question: The area of the triangle formed by the tangents from an external point (h, k) to the circle x² + y² = a² and the chord of contact, is -

• a) a(h² + k² - a² ) ^3/2
• b)

  

• c) None of these
• d)

  

Answer:

a(h² + k² - a² ) ^3/2

Question: The pole of the straight line 9x + y – 28 = 0 with respect to the circle 2x² + 2y² – 3x + 5y – 7 = 0, is

• a) (3,–1)
• b) (2, – 1)
• c) (2, 1)
• d) (3,1)

Answer: (3,–1)

Question: The equations of the tangents drawn from the origin to the circle x² + y² – 2rx – 2hy + h² = 0, are

(1) x = 0
(2) y = 0
(3) (h² – r²)x – 2rhy = 0
(4) (h² – r²)x + 2rhy = 0

• a) 1 and 3 are correct
• b) 1 and 2 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 3 are correct

Question: Consider the circle x² + y² – 10x – 6y + 30 = 0. Let O be the centre of the circle and tangent at A (7, 3) and B (5, 1) meet at C. Let S = 0 represents family of circles passings through A and B, then –

(1) Area of quadrilateral OACB = 4
(2) The smallest possible circle of the family of circles S = 0 is x² + y² – 12x – 4y + 38 = 0
(3) The coordinates of point C are (7, 1)
(4) The radical axis for the family of circles S = 0 is x + y = 10

• a) 1, 2 and 3 are correct
• b) 2 and 4 are correct
• c) 1 and 2 are correct
• d) 1 and 3 are correct

Answer: 1, 2 and 3 are correct

Question: A circle x² + y² = 1 cuts the line x + y = k and makes the chord with length l. The value of k is –

1. √6/2

2. -√6/2

3. √3/2

4. None of these

• a) 1 and 2 are correct
• b) 1 and 3 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 2 are correct

Question:

Statement-1 : Number of common tangents of x² + y² – 2x – 4y – 95 = 0 and x² + y² – 6x – 8y + 16 = 0 is zero.
Statement 2 : If C₁C₂ < | r₁ – r₂ |, then there will be no common tangent. (where C₁, C₂ are the centre and r₁, r₂ are radii of circles).

• a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
• b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• c) Statement-1 is False, Statement-2 is True.
• d) Statement-1 is True, Statement-2 is False.

Answer: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Question:

Statement-1 : The product of slopes of all the common tangents of circles S₁ = x² + y² – a² = 0 and S₂ = (x – 2a)² + (y – 2a)² – 4a² = 0 is 1.

Statement-2 : Slope of line joining centres of S₁ = x² + y² – a² = 0 and S² = (x – 2a)² + (y – 2a)² – 4a² = 0 is 1. Direct common tangents make equal angles with the line joining centres of the circles.

• a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
• b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• c) Statement-1 is False, Statement-2 is True.
• d) Statement-1 is True, Statement-2 is False.

Answer: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Question:

• a) Statement-1 is False, Statement-2 is True.
• b) Statement-1 is True, Statement-2 is False.
• c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: Statement-1 is False, Statement-2 is True.