JEE Mathematics Circles MCQs Set 03

Question. The angle between the pair of tangents from the point (1, 1/2) to the circle \( x^2 + y^2 + 4x + 2y – 4 = 0 \) is
(a) \( \cos^{-1}(4/5) \)
(b) \( \sin^{-1}(4/5) \)
(c) \( \sin^{-1}(3/5) \)
(d) None of the options
Answer: (b) \( \sin^{-1}(4/5) \)

Question. The chords of contact of the pair of tangents to the circle \( x^2 + y^2 = 1 \) drawn from any point on the line \( 2x + y = 4 \) pass through the point
(a) (1/2, 1/4)
(b) (1/4, 1/2)
(c) (1, 1/2)
(d) (1/2, 1)
Answer: (a) (1/2, 1/4)

Question. A foot of the normal from the point (4, 3) to a circle is (2, 1) and a diameter of the circle has the equation \( 2x – y = 2 \). Then the equation of the circle is
(a) \( x^2 + y^2 + 2x – 1 = 0 \)
(b) \( x^2 + y^2 - 2x – 1 = 0 \)
(c) \( x^2 + y^2 - 2y – 1 = 0 \)
(d) None of the options
Answer: (b) \( x^2 + y^2 - 2x – 1 = 0 \)

Question. The line \( \lambda x + \mu y = 1 \) is a normal to the circle \( 2x^2 + 2y^2 – 5x + 6y – 1 = 0 \) if
(a) \( 5\lambda - 6\mu = 2 \)
(b) \( 4 + 5\mu = 6\lambda \)
(c) \( 5\lambda - 6\mu = 4 \)
(d) None of the options
Answer: (c) \( 5\lambda - 6\mu = 4 \)

Question. The number of feet of normals from the point (7, -4) to the circle \( x^2 + y^2 = 5 \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The equation of a chord of the circle \( x^2 + y^2 – 4x = 0 \) which is bisected at the point (1, 1) is
(a) \( x + y = 2 \)
(b) \( 3x – y = 2 \)
(c) \( x – 2y + 1 = 0 \)
(d) \( x – y = 0 \)
Answer: (d) \( x – y = 0 \)

Question. Lines are drawn through the point P(-2, -3) to meet the circle \( x^2 + y^2 – 2x – 10y + 1 = 0 \). The length of the line segment PA, A being the point on the circle where the line meets the circle sat coincident points, is
(a) 16
(b) \( 4\sqrt{3} \)
(c) 48
(d) None of the options
Answer: (b) \( 4\sqrt{3} \)

Question. The equations of two circles are \( x^2 + y^2 + 2\lambda x + 5 = 0 \) and \( x^2 + y^2 + 2\lambda y + 5 = 0 \). P is any point on the line \( x - y = 0 \). If PA and PB are then lengths of the tangents from P to the two circles and PA = 3 then PB is equal to
(a) 1.5
(b) 6
(c) 3
(d) None of the options
Answer: (c) 3

Question. The common chord of the circle \( x^2 + y^2 + 6x + 8y – 7 = 0 \) and a circle passing through the origin, touching the line \( y = x \), always passes through the point
(a) (-1/2, 1/2)
(b) (1, 1)
(c) (1/2, 1/2)
(d) None of the options
Answer: (c) (1/2, 1/2)

Question. A tangent to the circle \( x^2 + y^2 = 1 \) through the point (0, 5) cuts the circle \( x^2 + y^2 = 4 \) at A and B. The tangents to the circle \( x^2 + y^2 = 4 \) at A and B meet at C. The coordinates of C are
(a) \( ( \frac{8}{5}\sqrt{6}, \frac{4}{5} ) \)
(b) \( ( \frac{8}{5}\sqrt{6}, -\frac{4}{5} ) \)
(c) \( ( -\frac{8}{5}\sqrt{6}, -\frac{4}{5} ) \)
(d) None of the options
Answer: (a) \( ( \frac{8}{5}\sqrt{6}, \frac{4}{5} ) \)

Question. If the common chord of the circles \( x^2 + (y - \lambda)^2 = 16 \) and \( x^2 + y^2 = 16 \) subtend a right angle at the origin then \( \lambda \) is equal to
(a) 4
(b) \( 4\sqrt{2} \)
(c) \( \pm 4\sqrt{2} \)
(d) 8
Answer: (c) \( \pm 4\sqrt{2} \)

Question. The equation of the smallest circle passing through the intersection of the line \( x + y = 1 \) and the circle \( x^2 + y^2 = 9 \) is
(a) \( x^2 + y^2 + x + y – 8 = 0 \)
(b) \( x^2 + y^2 – x – y – 8 = 0 \)
(c) \( x^2 + y^2 – x + y – 8 = 0 \)
(d) None of the options
Answer: (b) \( x^2 + y^2 – x – y – 8 = 0 \)

Question. The equation of a circle is \( x^2 + y^2 = 4 \). The centre of the smallest circle touching this circle and the line \( x + y = 5\sqrt{2} \) has the coordinates
(a) \( ( \frac{7}{2\sqrt{2}}, \frac{7}{2\sqrt{2}} ) \)
(b) \( ( 3/2, 3/2 ) \)
(c) \( ( -\frac{7}{2\sqrt{2}}, -\frac{7}{2\sqrt{2}} ) \)
(d) None of the options
Answer: (a) \( ( \frac{7}{2\sqrt{2}}, \frac{7}{2\sqrt{2}} ) \)

Question. The members of a family of circles are given by the equation \( 2(x^2 + y^2) + \lambda x – (1 + \lambda^2)y – 10 = 0 \). The number of circles belonging to the family that are cut orthogonally by the fixed circle \( x^2 + y^2 + 4x + 6y + 3 = 0 \) is
(a) 2
(b) 1
(c) 0
(d) None of the options
Answer: (a) 2

Question. The equation of the circle with the chord \( y = 2x \) of the circle \( x^2 + y^2 – 10x = 0 \) as its diameter is
(a) \( x^2 + y^2 – 2x – 4y – 5 = 0 \)
(b) \( x^2 + y^2 = 2x + 4y \)
(c) \( x^2 + y^2 = 4x + 2y \)
(d) None of the options
Answer: (b) \( x^2 + y^2 = 2x + 4y \)

Question. A circle of radius 2 touches the coordinate axes in the first quadrant. If the circle makes a complete rotation on the x-axis along the positive direction of the x-axis then the equation of the circle in the new position is
(a) \( x^2 + y^2 - 4(x + y) - 8\pi x + (2 + 4\pi)^2 = 0 \)
(b) \( x^2 + y^2 – 4x – 4y + (2 + 4\pi)^2 = 0 \)
(c) \( x^2 + y^2 - 8\pi x – 4y + (2 + 4\pi)^2 = 0 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 - 4(x + y) - 8\pi x + (2 + 4\pi)^2 = 0 \)

Question. A ray of light incident at the point (-2, -1) gets reflected from the tangent at (0, -1) to the circle \( x^2 + y^2 = 1 \). The reflected ray touches the circle. The equation of the line along which the incident ray moved is
(a) \( 4x – 3y + 11 = 0 \)
(b) \( 4x + 3y + 11 = 0 \)
(c) \( 3x + 4y + 11 = 0 \)
(d) None of the options
Answer: (b) \( 4x + 3y + 11 = 0 \)

Question. The locus of the centres of the circles for which one end of a diameter is (1, 1) while the other end is on the line \( x + y = 3 \) is
(a) \( x + y = 1 \)
(b) \( 2(x – y) = 5 \)
(c) \( 2x + 2y = 5 \)
(d) None of the options
Answer: (c) \( 2x + 2y = 5 \)

Question. The angle between a pair of tangents drawn from a point P to the curve \( x^2 + y^2 + 4x – 6y + 9\sin^2\alpha + 13\cos^2\alpha = 0 \) is \( 2\alpha \). The locus of P is
(a) \( x^2 + y^2 + 4x – 6y + 4 = 0 \)
(b) \( x^2 + y^2 + 4x – 6y – 9 = 0 \)
(c) \( x^2 + y^2 + 4x – 6y – 4 = 0 \)
(d) \( x^2 + y^2 + 4x – 6y + 9 = 0 \)
Answer: (d) \( x^2 + y^2 + 4x – 6y + 9 = 0 \)

Question. The point P moves in the plane of a regular hexagon such that the sum of the squares of its distances from the vertices of the hexagon is \( 6a^2 \). If the radius of the circumcircles of the hexagon is \( r(< a) \) then the locus of P is
(a) a pair of straight lines
(b) an ellipse
(c) a circle of radius \( \sqrt{a^2 – r^2} \)
(d) an ellipse of major axis a and minor axis r
Answer: (c) a circle of radius \( \sqrt{a^2 – r^2} \)

Question. The locus of the middle points of chords of length 4 of the circle \( x^2 + y^2 = 16 \) is
(a) a straight line
(b) a circle of radius 2
(c) a circle of radius \( 2\sqrt{3} \)
(d) an ellipse
Answer: (c) a circle of radius \( 2\sqrt{3} \)

Question. The equation of the locus of the middle point of a chord of the circle \( x^2 + y^2 = 2(x + y) \) such that the pair of lines joining the origin to the point of intersection of the chord and the circle are equally inclined to the x-axis is
(a) \( x + y = 2 \)
(b) \( x – y = 2 \)
(c) \( 2x – y = 1 \)
(d) None of the options
Answer: (a) \( x + y = 2 \)

Question. The locus of the centres of circles passing through the origin and intersecting the fixed circle \( x^2 + y^2 - 5x + 3y – 1 = 0 \) orthogonally is
(a) a straight line of the slope 3/5
(b) a circle
(c) a pair of straight lines
(d) None of the options
Answer: (d) None of the options

Question. The equation of a circle \( C_1 \) is \( x^2 + y^2 – 4x – 2y – 11 = 0 \). A circle \( C_2 \) of radius 1 rolls on the outside of the circle \( C_1 \). The locus of the centre of \( C_2 \) has the equation
(a) \( x^2 + y^2 – 4x – 2y – 20 = 0 \)
(b) \( x^2 + y^2 + 4x + 2y – 20 = 0 \)
(c) \( x^2 + y^2 – 3x – y – 11 = 0 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 – 4x – 2y – 20 = 0 \)

Question. Circles are drawn through the point (3, 0) to cut an intercept of length 6 units on the negative direction of the x-axis. The equation of the locus of their centres is
(a) the x-axis
(b) \( x – y = 0 \)
(c) the y-axis
(d) \( x + y = 0 \)
Answer: (c) the y-axis

Question. The locus of the centre of a circle touching the line \( x + 2y = 0 \) and \( x – 2y = 0 \) is
(a) \( xy = 0 \)
(b) \( x = 0 \)
(c) \( y = 0 \)
(d) None of the options
Answer: (a) \( xy = 0 \)

Question. The locus of a point from which the lengths of the tangents to the circles \( x^2 + y^2 = 4 \) and \( 2(x^2 + y^2) – 10x + 3y – 2 = 0 \) are equal to
(a) a straight line inclined at \( \pi/4 \) with the line joining the centres of the circles
(b) a circles
(c) an ellipse
(d) a straight line perpendicular to the line joining the centres of the circles
Answer: (d) a straight line perpendicular to the line joining the centres of the circles

Question. The locus of the centres of the circles passing through the intersection of the circles \( x^2 + y^2 = 1 \) and \( x^2 + y^2 - 2x + y = 0 \) is
(a) a line whose equation is \( x + 2y = 0 \)
(b) a line whose equation is \( 2x – y = 1 \)
(c) a circle
(d) a pair of lines
Answer: (a) a line whose equation is \( x + 2y = 0 \)

Choose the correct options. One or more options may be correct.

Question. The point of contact of tangent from the point (1, 2) to the circle \( x^2 + y^2 = 1 \) has the coordinates
(a) \( ( \frac{1 - 2\sqrt{19}}{5}, \frac{2 + \sqrt{19}}{5} ) \)
(b) \( ( \frac{1 - 2\sqrt{19}}{5}, \frac{2 - \sqrt{19}}{5} ) \)
(c) \( ( \frac{1 + 2\sqrt{19}}{5}, \frac{2 + \sqrt{19}}{5} ) \)
(d) \( ( \frac{1 + 2\sqrt{19}}{5}, \frac{2 - \sqrt{19}}{5} ) \)
Answer: (a) and (d)

Question. The equation of a circle \( C_1 \) is \( x^2 + y^2= 4 \). The locus of the intersection of orthogonal tangents to the circle is the curve \( C_2 \) and the locus of the intersection of perpendicular tangents to the curve \( C_2 \) is the curve \( C_3 \). Then
(a) \( C_3 \) is a circle
(b) the area enclosed by the curve \( C_3 \) is \( 8\pi \)
(c) \( C_2 \) and \( C_3 \) are circles with the same centre
(d) None of the options
Answer: (a) \( C_3 \) is a circle and (c) \( C_2 \) and \( C_3 \) are circles with the same centre

Question. A line parallel to the line \( x – 3y = 2 \) touches the circle \( x^2 + y^2 – 4x + 2y – 5 = 0 \) at the point
(a) (1, -4)
(b) (1, 2)
(c) (3, -4)
(d) (3, 2)
Answer: (b) (1, 2) and (c) (3, -4)

Question. A point on the line \( x = 3 \) from which the tangents drawn to the circle \( x^2 + y^2 = 8 \) are at right angles is
(a) \( (3, -\sqrt{7}) \)
(b) \( (3, \sqrt{23}) \)
(c) \( (3, \sqrt{7}) \)
(d) \( (3, -\sqrt{23}) \)
Answer: (a) \( (3, -\sqrt{7}) \) and (c) \( (3, \sqrt{7}) \)

Question. Tangents drawn from (2, 0) to the circle \( x^2 + y^2 = 1 \) touches the circle at A and B. Then
(a) \( A = ( \frac{1}{2}, \frac{\sqrt{3}}{2} ), B = ( \frac{1}{2}, -\frac{\sqrt{3}}{2} ) \)
(b) \( A = ( -\frac{1}{2}, \frac{\sqrt{3}}{2} ), B = ( \frac{1}{2}, -\frac{\sqrt{3}}{2} ) \)
(c) \( A = ( \frac{1}{2}, \frac{\sqrt{3}}{2} ), B = ( \frac{1}{2}, -\frac{\sqrt{3}}{2} ) \)
(d) \( A = ( \frac{1}{2}, -\frac{\sqrt{3}}{2} ), B = ( \frac{1}{2}, \frac{\sqrt{3}}{2} ) \)
Answer: (c) and (d)

Question. The equations of four circles are \( (x \pm a)^2 + (y \pm a)^2 = a^2 \). The radius of a circle touching all the four circles is
(a) \( (\sqrt{2} - 1)a \)
(b) \( 2\sqrt{2}a \)
(c) \( (\sqrt{2} + 1)a \)
(d) \( (2 + \sqrt{2})a \)
Answer: (a) \( (\sqrt{2} - 1)a \) and (c) \( (\sqrt{2} + 1)a \)

Question. Let a line through the point P(5, 10) cut the line \( l \) whose equation is \( x + 2y = 5 \), at Q and the circle C whose equation is \( x^2 + y^2 = 25 \), at A and B. Then
(a) P is the pole of the line \( l \) with respect to the circle C
(b) \( l \) is the polar of the point P with respect to the circle C
(c) PA, PQ, PB are in AP
(d) PQ is the HM of PA and PB
Answer: (a), (b), and (d)

Question. The equation of a circle is \( x^2 + y^2 – 4x + 2y – 4 = 0 \). With respect to the circle
(a) the pole of the line \( x – 2y + 5 = 0 \) is (1, 1)
(b) the chord of contact of real tangents from (1, 1) is the line \( x – 2y + 5 = 0 \)
(c) the polar of the point (1, 1) is \( x – 2y + 5 = 0 \)
(d) None of the options
Answer: (a) the pole of the line \( x – 2y + 5 = 0 \) is (1, 1) and (c) the polar of the point (1, 1) is \( x – 2y + 5 = 0 \)