Class 11 Mathematics Circles MCQs Set 02

Question. Number of circles drawn through two points is
(a) One
(b) Two
(c) Three
(d) Infinite
Answer: (d) Infinite

Question. If three lines are not concurrent and no two of them are parallel, number of circles drawn touching all the three lines
(a) 1
(b) 4
(c) 3
(d) Infinite
Answer: (b) 4

Question. Equation of circle passing through non-collinear points A, B, C is
(a) Equation of the circle on AB as diameter+ K (equation of AB) = 0
(b) Equation of the circle on AB as diameter+ K (equation of BC) = 0
(c) Equation of the circle on AB as diameter+ K (equation of CA) = 0
(d) Equation of the circle on BC as diameter+ K (equation of AC) = 0
Answer: (a) Equation of the circle on AB as diameter+ K (equation of AB) = 0

Question. The equation of the chord joining \( \alpha, \beta \) on the circle S = 0 is
(a) \( (x+g)\cos\left(\frac{\alpha+\beta}{2}\right) + (y+f)\sin\left(\frac{\alpha+\beta}{2}\right) = \cos\left(\frac{\alpha-\beta}{2}\right) \)
(b) \( (x+g)\cos\left(\frac{\alpha+\beta}{2}\right) + (y+f)\sin\left(\frac{\alpha+\beta}{2}\right) = r\cos\left(\frac{\alpha-\beta}{2}\right) \)
(c) \( (x-g)\cos\left(\frac{\alpha+\beta}{2}\right) + (y-f)\sin\left(\frac{\alpha+\beta}{2}\right) = r\cos\left(\frac{\alpha-\beta}{2}\right) \)
(d) \( (x+g)\cos\left(\frac{\alpha-\beta}{2}\right) + (y+f)\sin\left(\frac{\alpha-\beta}{2}\right) = r\cos\left(\frac{\alpha-\beta}{2}\right) \)
Answer: (b) \( (x+g)\cos\left(\frac{\alpha+\beta}{2}\right) + (y+f)\sin\left(\frac{\alpha+\beta}{2}\right) = r\cos\left(\frac{\alpha-\beta}{2}\right) \)

Question. If the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \) touches x - axis at \( (x_1, 0) \) then \( x^2 + 2gx + c = \)
(a) \( (x-x_1)^2 \)
(b) \( (x+x_1)^2 \)
(c) \( (y-y_1)^2 \)
(d) \( (y+y_1)^2 \)
Answer: (a) \( (x-x_1)^2 \)

Question. G.M of shortest and farthest distances from a point w.r.t a circle S = 0 is
(a) \( S_{11} \)
(b) \( \sqrt{S_{11}} \)
(c) \( S_{11}^2 \)
(d) \( 2r \)
Answer: (b) \( \sqrt{S_{11}} \)

Question. The locus of the point of intersection of the two tangents drawn to the circle \( x^2 + y^2 = a^2 \) which include an angle \( \alpha \) is
(a) \( x^2 + y^2 = a^2 \csc^2 \alpha/2 \)
(b) \( x^2 + y^2 = a^2 \cot^2 \alpha/2 \)
(c) \( x^2 + y^2 = a^2 \tan\alpha \)
(d) \( x^2 + y^2 = a^2 \tan \alpha/2 \)
Answer: (a) \( x^2 + y^2 = a^2 \csc^2 \alpha/2 \)

Question. A and B are two fixed points. The locus of P such that in \( \Delta PAB, \frac{\sin B}{\sin A} \) is a constant \( (\neq 1) \) is ________
(a) a circle
(b) pair of lines
(c) part of a circle
(d) line parallel to BC
Answer: (a) a circle

Question. The intercept made by the circle with centre (2, 3) and radius 6 on y-axis is
(a) \( 18\sqrt{2} \)
(b) \( 12\sqrt{2} \)
(c) \( 8\sqrt{2} \)
(d) \( 6\sqrt{2} \)
Answer: (c) \( 8\sqrt{2} \)

Question. The centre of the circle passing through origin and making intercepts 8 and -4 on x and y axes respectively is
(a) (4, -2)
(b) (-2, 4)
(c) (8, -4)
(d) both (a) & (b)
Answer: (a) (4, -2)

Question. 2x + y = 0 is the equation of a diameter of the circle which touches the lines 4x-3y+10=0 and 4x-3y-30=0. The centre and radius of the circle are
(a) (-2,1); 4
(b) (1,-2); 8
(c) (1,-2); 4
(d) (1,-2); 16
Answer: (c) (1,-2); 4

Question. The equation of the circle which has both the axes as its tangents and which passes through the point (1,2)
(a) \( x^2 + y^2 - 2x + 2y - 1 = 0 \)
(b) \( x^2 + y^2 - 2x + 2y + 1 = 0 \)
(c) \( x^2 + y^2 - 2x - 2y + 1 = 0 \)
(d) \( x^2 + y^2 - 2x - 2y - 1 = 0 \)
Answer: (c) \( x^2 + y^2 - 2x - 2y + 1 = 0 \)

Question. If a circle of radius 2 touches X-axis at (1,0) then its centre may be
(a) (1, 2) and (1, -2)
(b) (1, 2) and (2, 1)
(c) (-1, 2) and (1, -2)
(d) (-1, 2) and (-1, -2)
Answer: (a) (1, 2) and (1, -2)

Question. Equation of circles touching x-axis at the origin and the line 4x-3y+24=0 are
(a) \( x^2 + y^2 - 6y = 0, x^2 + y^2 + 24y = 0 \)
(b) \( x^2 + y^2 + 2y = 0, x^2 + y^2 - 18y = 0 \)
(c) \( x^2 + y^2 + 18x = 0, x^2 + y^2 - 8x = 0 \)
(d) \( x^2 + y^2 + 4x = 0, x^2 + y^2 - 16x = 0 \)
Answer: (a) \( x^2 + y^2 - 6y = 0, x^2 + y^2 + 24y = 0 \)

Question. Centre of the circle touching y-axis at (0,3) and making an intercept 2 units on positive X - axis is
(a) \( (\sqrt{10}, 3) \)
(b) \( (3, \sqrt{10}) \)
(c) \( (-\sqrt{10}, 3) \)
(d) \( (-\sqrt{10}, -3) \)
Answer: (a) \( (\sqrt{10}, 3) \)

PARAMETRIC EQUATIONS

Question. Parametric equation of the circle \( x^2 + y^2 = 16 \) are
(a) \( x = 4\cos\theta, y = 4\sin\theta \)
(b) \( x = 4\cos\theta, y = 4\tan\theta \)
(c) \( x = 4\cosh\theta, y = 4\sinh\theta \)
(d) \( x = 4\sec\theta, y = 4\tan\theta \)
Answer: (a) \( x = 4\cos\theta, y = 4\sin\theta \)

Question. Locus of the point (sec \( \theta \), tanh \( \theta \)) is
(a) \( x^2 + y^2 = 1 \)
(b) \( x^2 - y^2 = 1 \)
(c) \( x^2 + y^2 + 1 = 0 \)
(d) \( x^2 - y^2 = x + y \)
Answer: (a) \( x^2 + y^2 = 1 \)

Question. To the circle \( x^2 + y^2 + 8x - 4y + 4 = 0 \) tangent at the point \( \theta = \pi/4 \) is
(a) \( x + y + 2 - 4\sqrt{2} = 0 \)
(b) \( x - y + 2 - 4\sqrt{2} = 0 \)
(c) \( x + y + 2 + 4\sqrt{2} = 0 \)
(d) \( x - y - 2 - 4\sqrt{2} = 0 \)
Answer: (a) \( x + y + 2 - 4\sqrt{2} = 0 \)

CHORD OF CONTACT, POLE, POLAR, CONJUGATE POINTS AND LINES, INVERSE POINTS (POLE AND POLAR NOT FOR MAINS)

Question. The chord of contact of (2,1) w.r.t to the circle \( x^2 + y^2 + 4x + 4y + 1 = 0 \) is
(a) 2x+y+7=0
(b) 4x+3y+7=0
(c) 3x+4y+1=0
(d) not existing
Answer: (b) 4x+3y+7=0

Question. The polar of (2,-1) w.r.t \( x^2 + y^2 + 6x + 4y - 1 = 0 \) is 5x+y+k=0 then k =
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. Pole of 3x+5y+17=0 w.r.t the circle \( x^2 + y^2 + 4x + 6y + 9 = 0 \) is
(a) (-1,2)
(b) (1,2)
(c) (1,-2)
(d) (2,1)
Answer: (b) (1,2)

Question. If ax+by+c=0 is the polar of (1,1) w.r.t the circle \( x^2 + y^2 - 2x + 2y + 1 = 0 \) and H.C.F of a,b,c is equal to one then \( a^2 + b^2 + c^2 = \)
(a) 0
(b) 3
(c) 5
(d) 15
Answer: (c) 5

Question. If (1,1), (k,2) are conjugate points with respect to the circle \( x^2 + y^2 + 8x + 2y + 3 = 0 \), then k =
(a) -12
(b) -12/7
(c) -12/5
(d) -4
Answer: (c) -12/5

Question. If 3x+2y=3 and 2x+5y=1 are conjugate lines w.r.t the circle \( x^2 + y^2 = r^2 \) then \( r^2 = \)
(a) 3/16
(b) 16/3
(c) \( 4/\sqrt{3} \)
(d) 3/4
Answer: (a) 3/16

Question. The points (3,2), (2,3) w.r.t the circle \( x^2 + y^2 = 12 \) are
(a) extremities of a diameter
(b) conjugate points
(c) Inverse points
(d) lie on the circle
Answer: (b) conjugate points

Question. The length of chord of contact of the point (3,6) with respect to the circle \( x^2 + y^2 = 10 \) is
(a) \( 2\sqrt{70}/3 \)
(b) \( 6\sqrt{5} \)
(c) \( 3\sqrt{5} \)
(d) \( 12/\sqrt{5} \)
Answer: (a) \( 2\sqrt{70}/3 \)

Question. If the lines 2x + 3y - 4 = 0 and kx + 4y - 2 = 0 are conjugate with respect to the circle \( x^2 + y^2 = 4 \) then k - 1 =
(a) -5
(b) -6
(c) -4
(d) 5
Answer: (b) -6

Question. For the circle \( x^2 + y^2 - 2x - 4y - 4 = 0 \), the lines 2x+3y-1=0, 2x+y+5=0 are
(a) perpendicular tangents
(b) conjugate
(c) parallel tangents
(d) perpendicular chords
Answer: (b) conjugate

Question. The inverse point of (2 ,-3) w.r.t to circle \( x^2 + y^2 + 6x - 4y - 12 = 0 \) is
(a) \( (1/2, 1/2) \)
(b) \( (1/2, -1/2) \)
(c) \( (-1/2, 1/2) \)
(d) \( (-1/2, -1/2) \)
Answer: (d) \( (-1/2, -1/2) \)

Question. If the inverse of P(-3,5) w.r.t to a circle is (1,3) then polar of P w.r.t to the circle is
(a) x+2y=7
(b) 2x-2y+11=0
(c) 2x-y+1=0
(d) 2x-y-1=0
Answer: (a) x+2y=7

CHORD WITH MID POINT

Question. The equation of the chord of \( x^2 + y^2 - 4x + 6y + 3 = 0 \) whose mid point is (1,-2) is
(a) x+y+1=0
(b) 2x+3y+4=0
(c) x-y-3=0
(d) not existing
Answer: (c) x-y-3=0

Question. The pair of tangents from (2,1) to the circle \( x^2 + y^2 = 4 \) is
(a) \( 3x^2 + 4xy + 16x + 8y + 20 = 0 \)
(b) \( 3x^2 + 4xy + 16x - 8y + 20 = 0 \)
(c) \( 3x^2 + 4xy - 16x - 8y + 20 = 0 \)
(d) \( 3x^2 - 4xy - 16x + 8y - 20 = 0 \)
Answer: (c) \( 3x^2 + 4xy - 16x - 8y + 20 = 0 \)

Question. The pair of tangents from origin to the circle \( x^2 + y^2 + 4x + 2y + 3 = 0 \) is
(a) \( (2x+y)^2 = 3(x^2 + y^2) \)
(b) \( (4x+2y)^2 = 3(x^2 + y^2) \)
(c) \( (2x-y)^2 = 3(x^2 + y^2) \)
(d) not existing
Answer: (c) \( (2x-y)^2 = 3(x^2 + y^2) \)

CIRCLES-RELATIVE POSITIONS

Question. The circles \( x^2 + y^2 - 12x + 8y + 48 = 0 \), \( x^2 + y^2 - 4x + 2y - 4 = 0 \) are
(a) intersecting
(b) touching externally
(c) touching internally
(d) one is lying inside the other
Answer: (b) touching externally

Question. The circles \( x^2 + y^2 - 2x - 4y - 20 = 0 \), \( x^2 + y^2 + 4x - 2y + 4 = 0 \) are
(a) one lies outside the other
(b) one lies completely inside the other
(c) touch externally
(d) touch internally
Answer: (b) one lies completely inside the other

Question. The number of common tangents to \( x^2 + y^2 = 256, (x-3)^2 + (y-4)^2 = 121 \) is
(a) one
(b) two
(c) three
(d) zero
Answer: (a) one

Question. The internal centre of similitude of the circles \( x^2 + y^2 - 2x + 4y + 4 = 0 \), \( x^2 + y^2 + 4x - 2y + 1 = 0 \) divides the segment joining their centres in the ratio
(a) 1:2
(b) 2:1
(c) -1:2
(d) -2:1
Answer: (a) 1:2

Question. The external centre of similitude of the circle \( x^2 + y^2 - 12x + 8y + 48 = 0 \) and \( x^2 + y^2 - 4x + 2y - 4 = 0 \) divides the segment joining centres in the ratio
(a) 2:3
(b) 3:2
(c) -2:3
(d) -3:2
Answer: (c) -2:3

Question. If the two circles \( (x-1)^2 + (y-3)^2 = r^2 \) and \( x^2 + y^2 - 8x + 2y + 8 = 0 \) intersect in two distinct points, then
(a) r > 2
(b) 2 < r < 8
(c) r < 2
(d) r = 2
Answer: (b) 2 < r < 8

Question. If the distance between the centres of two circles of radii 3,4 is 25 then the length of the transverse common tangent is
(a) 24
(b) 12
(c) 26
(d) 13
Answer: (a) 24