JEE Mathematics Circles MCQs Set 07

Question. Circles are drawn touching the co-ordinate axis and having radius 2, then
(a) centre of these circles lie on the pair of lines \( y^2 - x^2 = 0 \)
(b) centre of these circles lie only on the line \( y = x \)
(c) Area of the quadrilateral whose vertices are centre of these circles is 16 sq. units.
(d) Area of the circle touching these four circles internally is \( 4\pi(3 + 2\sqrt{2}) \)
Answer: (a), (c), (d)
 

Question. For the circles \( S_1 \equiv x^2 + y^2 - 4x - 6y - 12 = 0 \) and \( S_2 \equiv x^2 + y^2 + 6x + 4y - 12 = 0 \) and the line \( L \equiv x + y = 0 \)
(a) L is common tangent of \( S_1 \) and \( S_2 \)
(b) L is common chord of \( S_1 \) and \( S_2 \)
(c) L is radical axis of \( S_1 \) & \( S_2 \)
(d) L is Perpendicular to the line joining the centre of \( S_1 \) & \( S_2 \)
Answer: (b), (c), (d)
 

Question. \( x^2 + y^2 + 6x = 0 \) and \( x^2 + y^2 - 2x = 0 \) are two circles, then
(a) They touch each other externally
(b) They touch each other internally
(c) Area of triangle formed by their common tangents is \( 3\sqrt{3} \) sq. units.
(d) Their common tangents do not form any triangle.
Answer: (a), (c)
 

Question. 3 circle of radii 1, 2 and 3 and centres at A, B and C respectively, touch each other. Another circle whose centre is P touches all these 3 circles externally, and has radius r. Also \( \angle PAB = \theta \) & \( \angle PAC = \alpha \).
(a) \( \cos\theta = \frac{3 - r}{3(1 + r)} \)
(b) \( \cos\alpha = \frac{2 - r}{2(1 + r)} \)
(c) \( r = \frac{6}{23} \)
(d) \( r = \frac{6}{\sqrt{23}} \)
Answer: (a), (b), (c)
 

Question. Slope of tangent to the circle \( (x - r)^2 + y^2 = r^2 \) at the point \( (x, y) \) lying on the circle is
(a) \( \frac{x}{y - r} \)
(b) \( \frac{r - x}{y} \)
(c) \( \frac{y^2 - x^2}{2xy} \)
(d) \( \frac{y^2 + x^2}{2xy} \)
Answer: (b), (c)
Solution:
\( (x - r)^2 + y^2 = r^2 \)
\( \implies \) \( x^2 + y^2 - 2xr = 0 \)
tangent at \( (x_1, y_1) \)
\( xx_1 + yy_1 - r (x + x_1) = 0 \)
\( (x_1 - r) x + yy_1 - rx_1 = 0 \)
slope \( m_T = \frac{r - x_1}{y_1} = \frac{r - x}{y} \)     (b)
\( \frac{r - x}{y} = \frac{2xr - 2x^2}{2xy} \)
\( = \frac{x^2 + y^2 - 2x^2}{2xy} = \frac{y^2 - x^2}{2xy} \)     (c)

Question. The centre(s) of the circle(s) passing through the points (0, 0), (1, 0) and touching the circle \( x^2 + y^2 = 9 \) is/are
(a) \( \left( \frac{3}{2}, \frac{1}{2} \right) \)
(b) \( \left( \frac{1}{2}, \frac{3}{2} \right) \)
(c) \( \left( \frac{1}{2}, 2^{1/2} \right) \)
(d) \( \left( \frac{1}{2}, -2^{1/2} \right) \)
Answer: (c), (d)
 

Question. Point M moved along the circle \( (x - 4)^2 + (y - 8)^2 = 20 \). Then it broke away from it and moving along a tangent to the circle cuts the x-axis at the point (-2, 0). The co-ordinates of the point on the circle at which the moving point broke away can be
(a) \( \left( -\frac{3}{5}, \frac{46}{5} \right) \)
(b) \( \left( -\frac{2}{5}, \frac{44}{5} \right) \)
(c) \( (6, 4) \)
(d) \( (3, 5) \)
Answer: (b), (c)
 

Question. Consider the circles \( x^2 + y^2 = 1 \) & \( x^2 + y^2 - 2x - 6y + 6 = 0 \). Then equation of a common tangent to the two circles is
(a) \( 4x - 3y - 5 = 0 \)
(b) \( x + 1 = 0 \)
(c) \( 3x + 4y - 5 = 0 \)
(d) \( y - 1 = 0 \)
Answer: (a), (b), (c), (d)
 

Question. The common chord of two intersecting circles \( C_1 \) and \( C_2 \) can be seen from their centres at the angles of 90^\circ and 60^\circ respectively. If the distance between their centres is equal to \(\sqrt{3} + 1\) then the radius of \( C_1 \) and \( C_2 \) are
(a) \( \sqrt{3} \) and 3
(b) \( \sqrt{2} \) and \( 2\sqrt{2} \)
(c) \( \sqrt{2} \) and 2
(d) \( 2\sqrt{2} \) and 4
Answer: (c) \( \sqrt{2} \) and 2
 

Question. A circle touches a straight line \( \ell x + my + n = 0 \) and cuts the circle \( x^2 + y^2 = 9 \) orthogonally, The locus of centres of such circles is
(a) \( (\ell x + my + n)^2 = (\ell^2 + m^2) (x^2 + y^2 - 9) \)
(b) \( (\ell x + my - n)^2 = (\ell^2 + m^2) (x^2 + y^2 - 9) \)
(c) \( (\ell x + my + n)^2 = (\ell^2 + m^2) (x^2 + y^2 + 9) \)
(d) None of the options
Answer: (a) \( (\ell x + my + n)^2 = (\ell^2 + m^2) (x^2 + y^2 - 9) \)
 

Question. The equation of the circle having the lines \( y^2 - 2y + 4x - 2xy = 0 \) as its normals & passing through the point (2, 1) is
(a) \( x^2 + y^2 - 2x - 4y + 3 = 0 \)
(b) \( x^2 + y^2 - 2x + 4y - 5 = 0 \)
(c) \( x^2 + y^2 + 2x + 4y - 13 = 0 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 - 2x - 4y + 3 = 0 \)
 

Question. A circle is drawn touching the x-axis and centre at the point which is the reflection of (a, b) in the line y - x = 0. The equation of the circle is
(a) \( x^2 + y^2 - 2bx - 2ay + a^2 = 0 \)
(b) \( x^2 + y^2 - 2bx - 2ay + b^2 = 0 \)
(c) \( x^2 + y^2 - 2ax - 2by + b^2 = 0 \)
(d) \( x^2 + y^2 - 2ax - 2by + a^2 = 0 \)
Answer: (a) \( x^2 + y^2 - 2bx - 2ay + a^2 = 0 \)
 

Question. The length of the common chord of circles \( x^2 + y^2 - 6x - 16 = 0 \) and \( x^2 + y^2 - 8y - 9 = 0 \) is
(a) \( 10\sqrt{3} \)
(b) \( 5\sqrt{3} \)
(c) \( 5\sqrt{3}/2 \)
(d) None of the options
Answer: (b) \( 5\sqrt{3} \)
 

Question. The number of common tangents of the circles \( x^2 + y^2 - 2x - 1 = 0 \) and \( x^2 + y^2 - 2y - 7 = 0 \)
(a) 1
(b) 3
(c) 2
(d) 4
Answer: (a) 1
 

Question. The point from which the tangents to the circles
\( x^2 + y^2 - 8x + 40 = 0 \)
\( 5x^2 + 5y^2 - 25x + 80 = 0 \)
\( x^2 + y^2 - 8x + 16y + 160 = 0 \)
are equal in length is
(a) \( (8, \frac{15}{2}) \)
(b) \( (-8, \frac{15}{2}) \)
(c) \( (8, -\frac{15}{2}) \)
(d) None of the options
Answer: (c) \( (8, -\frac{15}{2}) \)
 

Question. If the circle \( x^2 + y^2 = 9 \) touches the circle \( x^2 + y^2 + 6y + c = 0 \), then c is equal to
(a) -27
(b) 36
(c) -36
(d) 27
Answer: (a) -27
 

Question. If the two circles, \( x^2 + y^2 + 2g_1x + 2f_1y = 0 \) and \( x^2 + y^2 + 2g_2x + 2f_2y = 0 \) touches each other, then
(a) \( f_1g_1 = f_2g_2 \)
(b) \( \frac{f_1}{g_1} = \frac{f_2}{g_2} \)
(c) \( f_1f_2 = g_1g_2 \)
(d) None of the options
Answer: (b) \( \frac{f_1}{g_1} = \frac{f_2}{g_2} \)
 

Question. If \( (a, \frac{1}{a}) \), \( (b, \frac{1}{b}) \), \( (c, \frac{1}{c}) \) & \( (d, \frac{1}{d}) \) are four distinct points on a circle of radius 4 units then, abcd =
(a) 4
(b) 1/4
(c) 1
(d) 16
Answer: (c) 1
 

Question. The tangent from the point of intersection of the lines \( 2x - 3y + 1 = 0 \) and \( 3x - 2y - 1 = 0 \) to the circle \( x^2 + y^2 + 2x - 4y = 0 \) is
(a) \( x + 2y = 0, x - 2y + 1 = 0 \)
(b) \( 2x - y - 1 = 0 \)
(c) \( y = x, y = 3x - 2 \)
(d) \( 2x + y + 1 = 0 \)
Answer: (b) \( 2x - y - 1 = 0 \)
 

Question. What is the length of shortest path by which one can go from (-2, 0) to (2, 0) without entering the interior of circle, \( x^2 + y^2 = 1 \)
(a) \( 2\sqrt{3} \)
(b) \( \sqrt{3} + \frac{2\pi}{3} \)
(c) \( 2\sqrt{3} + \frac{\pi}{3} \)
(d) None of the options
Answer: (c) \( 2\sqrt{3} + \frac{\pi}{3} \)
 

Question. Three equal circles each of radius r touch one another. The radius of the circle touching all the three given circle internally is
(a) \( (2 + \sqrt{3})r \)
(b) \( \frac{2 + \sqrt{3}}{\sqrt{3}}r \)
(c) \( \frac{2 - \sqrt{3}}{\sqrt{3}}r \)
(d) \( (2 - \sqrt{3})r \)
Answer: (b) \( \frac{2 + \sqrt{3}}{\sqrt{3}}r \)
 

Question. (a) The triangle PQR is inscribed in the circle, \( x^2+y^2=25 \). If Q and R have co-ordinates (3, 4) & (-4, 3) respectively, then \( \angle QPR \) is equal to
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{6} \)
Answer: (c) \( \frac{\pi}{4} \)

Question. (b) If the circles, \( x^2 + y^2 + 2x + 2ky + 6 = 0 \) & \( x^2 + y^2 + 2ky + k = 0 \) intersect orthogonally, then 'k' is
(a) 2 or \( -\frac{3}{2} \)
(b) -2 or \( -\frac{3}{2} \)
(c) 2 or \( \frac{3}{2} \)
(d) -2 or \( \frac{3}{2} \)
Answer: (a) 2 or \( -\frac{3}{2} \)

Question. (a) Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle then 2r equals.
(a) \( \sqrt{PQ \cdot RS} \)
(b) \( \frac{PQ + RS}{2} \)
(c) \( \frac{2PQ \cdot RS}{PQ + RS} \)
(d) \( \sqrt{\frac{(PQ)^2 + (RS)^2}{2}} \)
Answer: (a) \( \sqrt{PQ \cdot RS} \)

Question. (a) If the tangent at the point P on the circle \( x^2 + y^2 + 6x +6y = 2 \) meets the straight line \( 5x - 2y + 6 = 0 \) at a point Q on the y-axis, then the length of PQ is
(a) 4
(b) \( 2\sqrt{5} \)
(c) 5
(d) \( 3\sqrt{5} \)
Answer: (c) 5

Question. (b) If \( a > 2b > 0 \) then the positive value of m for which \( y = mx - b\sqrt{1+m^2} \) is a common tangent to \( x^2 + y^2 = b^2 \) and \( (x - a)^2 + y^2 = b^2 \) is
(a) \( \frac{2b}{\sqrt{a^2 - 4b^2}} \)
(b) \( \frac{\sqrt{a^2 - 4b^2}}{2b} \)
(c) \( \frac{2b}{a - 2b} \)
(d) \( \frac{b}{a - 2b} \)
Answer: (a) \( \frac{2b}{\sqrt{a^2 - 4b^2}} \)

Question. The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle \( x^2 + y^2 - 2x - 6y + 6 = 0 \)
(a) 1
(b) 2
(c) 3
(d) \( \sqrt{3} \)
Answer: (c) 3

Question. A circle is given by \( x^2 + (y - 1)^2 = 1 \), another circle C touches it externally and also the x-axis, then the locus of its centre is
(a) \( \{(x,y) : x^2 = 4y\} \cup \{(x,y) : y \le 0\} \)
(b) \( \{(x,y) : x^2 + (y - 1)^2 = 4\} \cup \{(x,y) : y \le 0\} \)
(c) \( \{(x,y) : x^2 = y\} \cup \{(0,y) : y \le 0\} \)
(d) \( \{(x,y) : x^2 = 4y\} \cup \{(0,y) : y \le 0\} \)
Answer: (d) \( \{(x,y) : x^2 = 4y\} \cup \{(0,y) : y \le 0\} \)

Question. (a) Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is
(a) 3
(b) 2
(c) 3/2
(d) 1
Answer: (b) 2

Question. (b) Tangents are drawn from the point (17, 7) to the circle \( x^2 + y^2 = 169 \).
Statement-I : The tangents are mutually perpendicular. because
Statement-II : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is \( x^2 + y^2 = 338 \).
(a) Statement-I is true, statement-II is true; statement-II is correct explanation for statement-I
(b) Statement-I is true, statement-II is true; statement-II is NOT correct explanation for statement-I
(c) Statement-I is true, Statement-II is False
(d) Statement-I is False, Statement-II is True
Answer: (a) Statement-I is true, statement-II is true; statement-II is correct explanation for statement-I

Question. (a) Consider the two curves \( C_1 : y^2 = 4x \) ; \( C_2 : x^2 + y^2 - 6x + 1 = 0 \). Then,
(a) \( C_1 \) and \( C_2 \) touch each other only at one point
(b) \( C_1 \) and \( C_2 \) touch each other exactly at two points
(c) \( C_1 \) and \( C_2 \) intersect (but do not touch) at exactly two points
(d) \( C_1 \) and \( C_2 \) neither intersect nor touch each other
Answer: (b) \( C_1 \) and \( C_2 \) touch each other exactly at two points

Question. (b) Consider, \( L_1 : 2x + 3y + p - 3 = 0 \) ; \( L_2 : 2x + 3y + p + 3 = 0 \), where p is a real number, and \( C : x^2 + y^2 + 6x - 10y + 30 = 0 \)
Statement-I : If line \( L_1 \) is a chord of circle C, then line \( L_2 \) is not always a diameter of circle C. and
Statement-II : If line \( L_1 \) is a diameter of circle C, then line \( L_2 \) is not a chord of circle C.
(a) Statement-I is true, statement-II is true; statement-II is correct explanation for statement-I
(b) Statement-I is true, statement-II is true; statement-II is NOT correct explanation for statement-I
(c) Statement-I is true, Statement-II is False
(d) Statement-I is False, Statement-II is True
Answer: (c) Statement-I is true, Statement-II is False

Comprehension (3 questions together) :
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP and D, E, F respectively. The line PQ is given by the equation \( \sqrt{3}x + y - 6 = 0 \) and the point D is \( \left(\frac{3\sqrt{3}}{2}, \frac{3}{2}\right) \). Further, it is given that the origin and the centre of C are on the same side of the line PQ.

Question. (i) The equation of circle C is
(a) \( (x - 2\sqrt{3})^2 + (y - 1)^2 = 1 \)
(b) \( (x - 2\sqrt{3})^2 + \left(y + \frac{1}{2}\right)^2 = 1 \)
(c) \( (x - \sqrt{3})^2 + (y + 1)^2 = 1 \)
(d) \( (x - \sqrt{3})^2 + (y - 1)^2 = 1 \)
Answer: (d) \( (x - \sqrt{3})^2 + (y - 1)^2 = 1 \)

Question. (ii) Points E and F are given by
(a) \( \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right), (\sqrt{3}, 0) \)
(b) \( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right), (\sqrt{3}, 0) \)
(c) \( \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right), \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \)
(d) \( \left(\frac{3}{2}, \frac{\sqrt{3}}{2}\right), \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \)
Answer: (a) \( \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right), (\sqrt{3}, 0) \)

Question. (iii) Equations of the sides RP, RQ are
(a) \( y = \frac{2}{\sqrt{3}}x + 1, y = -\frac{2}{\sqrt{3}}x - 1 \)
(b) \( y = \frac{1}{\sqrt{3}}x, y = 0 \)
(c) \( y = \frac{\sqrt{3}}{2}x + 1, y = -\frac{\sqrt{3}}{2}x - 1 \)
(d) \( y = \sqrt{3}x, y = 0 \)
Answer: (d) \( y = \sqrt{3}x, y = 0 \)

Question. (a) Tangents drawn from the point P(1, 8) to the circle \( x^2 + y^2 - 6x - 4y - 11 = 0 \) touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is
(a) \( x^2 + y^2 + 4x - 6y + 19 = 0 \)
(b) \( x^2 + y^2 - 4x - 10y + 19 = 0 \)
(c) \( x^2 + y^2 - 2x + 6y - 29 = 0 \)
(d) \( x^2 + y^2 - 6x - 4y + 19 = 0 \)
Answer: (b) \( x^2 + y^2 - 4x - 10y + 19 = 0 \)

Question. The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point
(a) \( \left(-\frac{3}{2}, 0\right) \)
(b) \( \left(-\frac{5}{2}, 2\right) \)
(c) \( \left(-\frac{3}{2}, \frac{5}{2}\right) \)
(d) \( (-4, 0) \)
Answer: (d) \( (-4, 0) \)

Question. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line \( 4x - 5y = 20 \) to the circle \( x^2 + y^2 = 9 \) is
(a) \( 20(x^2 + y^2) - 36x + 45y = 0 \)
(b) \( 20(x^2 + y^2) + 36x - 45y = 0 \)
(c) \( 36(x^2 + y^2) - 20x + 45y = 0 \)
(d) \( 36(x^2 + y^2) + 20x - 45y = 0 \)
Answer: (a) \( 20(x^2 + y^2) - 36x + 45y = 0 \)

Paragraph for Question Nos. 16 to 17
A tangent PT is drawn to the circle \( x^2 + y^2 = 4 \) at the point \( P(\sqrt{3}, 1) \). A straight line L, perpendicular to PT is a tangent to the circle \( (x - 3)^2 + y^2 = 1 \).

Question. A possible equation of L is
(a) \( x - \sqrt{3}y = 1 \)
(b) \( x + \sqrt{3}y = 1 \)
(c) \( x - \sqrt{3}y = -1 \)
(d) \( x + \sqrt{3}y = 5 \)
Answer: (a) \( x - \sqrt{3}y = 1 \)

Question. A common tangent of the two circles is
(a) \( x = 4 \)
(b) \( y = 2 \)
(c) \( x + \sqrt{3}y = 4 \)
(d) \( x + 2\sqrt{2}y = 6 \)
Answer: (d) \( x + 2\sqrt{2}y = 6 \)