JEE Mathematics Circles MCQs Set 06

Question. The lines \( 2x - 3y = 5 \) and \( 3x - 4y = 7 \) are diameters of a circle of area 154 sq. units. The equation of the circle is
(a) \( x^2 + y^2 - 2x - 2y = 47 \)
(b) \( x^2 + y^2 - 2x - 2y = 62 \)
(c) \( x^2 + y^2 - 2x + 2y = 47 \)
(d) \( x^2 + y^2 - 2x + 2y = 62 \)
Answer: (c) \( x^2 + y^2 - 2x + 2y = 47 \)
 

Question. If a be the radius of a circle which touches x-axis at the origin, then its equation is
(a) \( x^2 + y^2 + ax = 0 \)
(b) \( x^2 + y^2 \pm 2ya = 0 \)
(c) \( x^2 + y^2 \pm 2xa = 0 \)
(d) \( x^2 + y^2 + ya = 0 \)
Answer: (b) \( x^2 + y^2 \pm 2ya = 0 \)

Question. The equation of the circle which touches the axis of y at the origin and passes through (3, 4) is
(a) \( 4(x^2 + y^2) - 25x = 0 \)
(b) \( 3(x^2 + y^2) - 25x = 0 \)
(c) \( 2(x^2 + y^2) - 3x = 0 \)
(d) \( 4(x^2 + y^2) - 25x + 10 = 0 \)
Answer: (b) \( 3(x^2 + y^2) - 25x = 0 \)

Question. The equation of the circle passing through (3, 6) and whose centre is (2, -1) is
(a) \( x^2 + y^2 - 4x + 2y = 45 \)
(b) \( x^2 + y^2 - 4x - 2y + 45 = 0 \)
(c) \( x^2 + y^2 + 4x - 2y = 45 \)
(d) \( x^2 + y^2 - 4x + 2y + 45 = 0 \)
Answer: (a) \( x^2 + y^2 - 4x + 2y = 45 \)

Question. The equation to the circle whose radius is 4 and which touches the negative x-axis at a distance 3 units from the origin is
(a) \( x^2 + y^2 - 6x + 8y - 9 = 0 \)
(b) \( x^2 + y^2 \pm 6x - 8y + 9 = 0 \)
(c) \( x^2 + y^2 + 6x \pm 8y + 9 = 0 \)
(d) \( x^2 + y^2 \pm 6x - 8y - 9 = 0 \)
Answer: (c) \( x^2 + y^2 + 6x \pm 8y + 9 = 0 \)

Question. The equation of a circle which passes through the three points (3, 0) (1, -6), (4, -1) is
(a) \( 2x^2 + 2y^2 + 5x - 11y + 3 = 0 \)
(b) \( x^2 + y^2 - 5x + 11y - 3 = 0 \)
(c) \( x^2 + y^2 + 5x - 11y + 3 = 0 \)
(d) \( 2x^2 + 2y^2 - 5x + 11y - 3 = 0 \)
Answer: (d) \( 2x^2 + 2y^2 - 5x + 11y - 3 = 0 \)

Question. y = \(\sqrt{3}x + c_1\) & y = \(\sqrt{3}x + c_2\) are two parallel tangents of a circle of radius 2 units, then |c_1 - c_2| is equal to
(a) 8
(b) 4
(c) 2
(d) 1
Answer: (a) 8

Question. Number of different circles that can be drawn touching 3 lines, no two of which are parallel and they are neither coincident nor concurrent, are
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. B and C are fixed point having co-ordinates (3, 0) and (-3, 0) respectively. If the vertical angle BAC is 90^\circ, then the locus of the centroid of the \(\Delta ABC\) has the equation
(a) \( x^2 + y^2 = 1 \)
(b) \( x^2 + y^2 = 2 \)
(c) \( 9(x^2 + y^2) = 1 \)
(d) \( 9(x^2 + y^2) = 4 \)
Answer: (a) \( x^2 + y^2 = 1 \)

Question. If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes at A and B then the locus of the centroid of the triangle OAB is
(a) \( x^2 + y^2 = (2k)^2 \)
(b) \( x^2 + y^2 = (3k)^2 \)
(c) \( x^2 + y^2 = (4k)^2 \)
(d) \( x^2 + y^2 = (6k)^2 \)
Answer: (a) \( x^2 + y^2 = (2k)^2 \)

Question. The area of an equilateral triangle inscribed in the circle \( x^2 + y^2 - 2x = 0 \) is
(a) \( \frac{3\sqrt{3}}{2} \)
(b) \( \frac{3\sqrt{3}}{4} \)
(c) \( \frac{3\sqrt{3}}{8} \)
(d) None of the options
Answer: (b) \( \frac{3\sqrt{3}}{4} \)

Question. The length of intercept on y-axis, by a circle whose diameter is the line joining the points (-4,3) and (12,-1) is
(a) \( 3\sqrt{2} \)
(b) \( \sqrt{13} \)
(c) \( 4\sqrt{13} \)
(d) None of the options
Answer: (c) \( 4\sqrt{13} \)

Question. The gradient of the tangent line at the point \( (a \cos \alpha, a \sin \alpha) \) to the circle \( x^2 + y^2 = a^2 \), is
(a) \( \tan(\pi - \alpha) \)
(b) \( \tan \alpha \)
(c) \( \cot \alpha \)
(d) \( -\cot \alpha \)
Answer: (d) \( -\cot \alpha \)

Question. \( \ell x + my + n = 0 \) is a tangent line to the circle \( x^2 + y^2 = r^2 \), if
(a) \( \ell^2 + m^2 = n^2 r^2 \)
(b) \( \ell^2 + m^2 = n^2 + r^2 \)
(c) \( n^2 = r^2(\ell^2 + m^2) \)
(d) None of the options
Answer: (c) \( n^2 = r^2(\ell^2 + m^2) \)

Question. If y=c is a tangent to the circle \( x^2 + y^2 - 2x + 2y - 2 = 0 \) at (1, 1), then the value of c is
(a) 1
(b) 2
(c) -1
(d) -2
Answer: (a) 1

Question. Line 3x + 4y = 25 touches the circle \( x^2 + y^2 = 25 \) at the point
(a) (4, 3)
(b) (3, 4)
(c) (-3, -4)
(d) None of the options
Answer: (b) (3, 4)

Question. The equations of the tangents drawn from the point (0, 1) to the circle \( x^2 + y^2 - 2x + 4y = 0 \) are
(a) \( 2x - y + 1 = 0, x + 2y - 2 = 0 \)
(b) \( 2x - y - 1 = 0, x + 2y - 2 = 0 \)
(c) \( 2x - y + 1 = 0, x + 2y + 2 = 0 \)
(d) \( 2x - y - 1 = 0, x + 2y + 2 = 0 \)
Answer: (a) \( 2x - y + 1 = 0, x + 2y - 2 = 0 \)

Question. The greatest distance of the point P(10, 7) from the circle \( x^2 + y^2 - 4x - 2y - 20 = 0 \) is
(a) 5
(b) 15
(c) 10
(d) None of the options
Answer: (b) 15

Question. The equation of the normal to the circle \( x^2 + y^2 = 9 \) at the point \( (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}) \) is
(a) \( x - y = \frac{\sqrt{2}}{3} \)
(b) \( x + y = 0 \)
(c) \( x - y = 0 \)
(d) None of the options
Answer: (c) \( x - y = 0 \)

Question. The parametric coordinates of any point on the circle \( x^2 + y^2 - 4x - 4y = 0 \) are
(a) \( (-2 + 2\cos\alpha, -2 + 2\sin\alpha) \)
(b) \( (2 + 2\cos\alpha, 2 + 2\sin\alpha) \)
(c) \( (2 + 2\sqrt{2}\cos\alpha, 2 + 2\sqrt{2}\sin\alpha) \)
(d) None of the options
Answer: (c) \( (2 + 2\sqrt{2}\cos\alpha, 2 + 2\sqrt{2}\sin\alpha) \)

Question. The length of the tangent drawn from the point (2, 3) to the circles \( 2(x^2 + y^2) - 7x + 9y - 11 = 0 \).
(a) 18
(b) 14
(c) \( \sqrt{14} \)
(d) \( \sqrt{28} \)
Answer: (c) \( \sqrt{14} \)

Question. A pair of tangents are drawn from the origin to the circle \( x^2 + y^2 + 20(x + y) + 20 = 0 \). The equation of the pair of tangents is
(a) \( x^2 + y^2 + 5xy = 0 \)
(b) \( x^2 + y^2 + 10xy = 0 \)
(c) \( 2x^2 + 2y^2 + 5xy = 0 \)
(d) \( 2x^2 + 2y^2 - 5xy = 0 \)
Answer: (c) \( 2x^2 + 2y^2 + 5xy = 0 \)

Question. Tangents are drawn from (4, 4) to the circle \( x^2 + y^2 - 2x - 2y - 7 = 0 \) to meet the circle at A and B. The length of the chord AB is
(a) \( 2\sqrt{3} \)
(b) \( 3\sqrt{2} \)
(c) \( 2\sqrt{6} \)
(d) \( 6\sqrt{2} \)
Answer: (b) \( 3\sqrt{2} \)

Question. The angle between the two tangents from the origin to the circle \( (x - 7)^2 + (y + 1)^2 = 25 \) equals
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) None of the options
Answer: (a) \( \frac{\pi}{2} \)

Question. Pair of tangents are drawn from every point on the line 3x + 4y = 12 on the circle \( x^2 + y^2 = 4 \). Their variable chord of contact always passes through a fixed point whose co-ordinates are
(a) \( (\frac{4}{3}, \frac{3}{4}) \)
(b) \( (\frac{3}{4}, \frac{3}{4}) \)
(c) \( (1, 1) \)
(d) \( (1, \frac{4}{3}) \)
Answer: (d) \( (1, \frac{4}{3}) \)

Question. The locus of the mid-points of the chords of the circle \( x^2 + y^2 - 2x - 4y - 11 = 0 \) which subtend 60^\circ at the centre is
(a) \( x^2 + y^2 - 4x - 2y - 7 = 0 \)
(b) \( x^2 + y^2 + 4x + 2y - 7 = 0 \)
(c) \( x^2 + y^2 - 2x - 4y - 7 = 0 \)
(d) \( x^2 + y^2 + 2x + 4y + 7 = 0 \)
Answer: (c) \( x^2 + y^2 - 2x - 4y - 7 = 0 \)

Question. The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord 5x + 2y = 16 is
(a) \( 2x - 5y + 11 = 0 \)
(b) \( 2x + 5y - 11 = 0 \)
(c) \( 2x + 5y + 11 = 0 \)
(d) None of the options
Answer: (a) \( 2x - 5y + 11 = 0 \)

Question. The locus of the centre of a circle which touches externally the circle, \( x^2 + y^2 - 6x - 6y + 14 = 0 \) and also touches the y-axis is given by the equation
(a) \( x^2 - 6x - 10y + 14 = 0 \)
(b) \( x^2 - 10x - 6y + 14 = 0 \)
(c) \( y^2 - 6x - 10y + 14 = 0 \)
(d) \( y^2 - 10x - 6y + 14 = 0 \)
Answer: (d) \( y^2 - 10x - 6y + 14 = 0 \)
 

Question. In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals to
(a) \( \frac{AB \cdot AD}{\sqrt{AB^2 + AD^2}} \)
(b) \( \frac{AB \cdot AD}{AB + AD} \)
(c) \( \sqrt{AB \cdot AD} \)
(d) \( \frac{AB \cdot AD}{\sqrt{AB^2 - AD^2}} \)
Answer: (d) \( \frac{AB \cdot AD}{\sqrt{AB^2 - AD^2}} \)

Question. The locus of the centers of the circles which cut the circles \( x^2 + y^2 + 4x - 6y + 9 = 0 \) and \( x^2 + y^2 - 5x + 4y - 2 = 0 \) orthogonally is
(a) \( 9x + 10y - 7 = 0 \)
(b) \( x - y + 2 = 0 \)
(c) \( 9x - 10y + 11 = 0 \)
(d) \( 9x + 10y + 7 = 0 \)
Answer: (c) \( 9x - 10y + 11 = 0 \)

Question. Tangents are drawn to the circle \( x^2 + y^2 = 1 \) at the points where it is met by the circles. \( x^2 + y^2 - (\lambda + 6)x + (8 - 2\lambda)y - 3 = 0 \), \( \lambda \) being the variable. The locus of the point of intersection of these tangents is
(a) \( 2x - y + 10 = 0 \)
(b) \( x + 2y - 10 = 0 \)
(c) \( x - 2y + 10 = 0 \)
(d) \( 2x + y - 10 = 0 \)
Answer: (a) \( 2x - y + 10 = 0 \)
 

Question. The circle passing through the distinct points (1, t), (t, 1) & (t, t) for all values of 't'. passes through the point
(a) (-1, -1)
(b) (-1, 1)
(c) (1, -1)
(d) (1, 1)
Answer: (d) (1, 1)
 

Question. AB is a diameter of a circle. CD is a chord parallel to AB and 2CD = AB. The tangent at B meets the line AC produced at E then AE is equal to
(a) AB
(b) \( \sqrt{2}AB \)
(c) \( 2\sqrt{2}AB \)
(d) 2AB
Answer: (d) 2AB
 

Question. The locus of the mid points of the chords of the circle \( x^2 + y^2 - ax - by = 0 \) which subtend a right angle at \( (\frac{a}{2}, \frac{b}{2}) \) is
(a) \( ax + by = 0 \)
(b) \( ax + by = a^2 + b^2 \)
(c) \( x^2 + y^2 - ax - by + \frac{a^2 + b^2}{8} = 0 \)
(d) \( x^2 + y^2 - ax - by - \frac{a^2 + b^2}{8} = 0 \)
Answer: (c) \( x^2 + y^2 - ax - by + \frac{a^2 + b^2}{8} = 0 \)
 

Question. A variable circle is drawn to touch the x-axis at the origin. The locus of the pole of the straight line \( \ell x + my + n = 0 \) w.r.t. the variable circle has the equation
(a) \( x(my - n) - \ell y^2 = 0 \)
(b) \( x(my + n) - \ell y^2 = 0 \)
(c) \( x(my - n) + \ell y^2 = 0 \)
(d) None of the options
Answer: (a) \( x(my - n) - \ell y^2 = 0 \)
 

Question. (6, 0), (0, 6) and (7, 7) are the vertices of a triangle. The circle inscribed in the triangle has the equation
(a) \( x^2 + y^2 - 9x + 9y + 36 = 0 \)
(b) \( x^2 + y^2 - 9x - 9y + 36 = 0 \)
(c) \( x^2 + y^2 + 9x - 9y + 36 = 0 \)
(d) \( x^2 + y^2 - 9x - 9y - 36 = 0 \)
Answer: (b) \( x^2 + y^2 - 9x - 9y + 36 = 0 \)
 

Question. A circle is inscribed into a rhombus ABCD with one angle 60^\circ. The distance from the centre of the circle to the nearest vertex is equal to 1. If P is any point of the circle, then |PA|^2 + |PB|^2 + |PC|^2 + |PD|^2 is equal to
(a) 12
(b) 11
(c) 9
(d) None of the options
Answer: (b) 11
 

Question. Number of points (x, y) having integral coordinates satisfying the condition \( x^2 + y^2 < 25 \) is
(a) 69
(b) 80
(c) 81
(d) 77
Answer: (a) 69
 

Question. If \( a^2 + b^2 = 1 \), \( m^2 + n^2 = 1 \), then
(a) \( |am + bn| \leq 1 \)
(b) \( |am - bn| \geq 1 \)
(c) \( |am + bn| \geq 1 \)
(d) None of the options
Answer: (a) \( |am + bn| \leq 1 \)
 

Question. The distance between the chords of contact of tangents to the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \) from the origin and from the point (g, f) is
(a) \( \frac{\sqrt{g^2 + f^2}}{2} \)
(b) \( \frac{g^2 + f^2 - c}{2} \)
(c) \( \frac{g^2 + f^2 - c}{2\sqrt{g^2 + f^2}} \)
(d) \( \frac{g^2 + f^2 + c}{2\sqrt{g^2 + f^2}} \)
Answer: (c) \( \frac{g^2 + f^2 - c}{2\sqrt{g^2 + f^2}} \)