JEE Mathematics Circles MCQs Set A

Question. If the equation of a circle is \( ax^2 + (2a – 3)y^2 – 4x – 1 = 0 \) then its centre is
(a) \( (2, 0) \)
(b) \( (2/3, 0) \)
(c) \( (-2/3, 0) \)
(d) None of the options
Answer: (b) \( (2/3, 0) \)

Question. If \( 2x^2 + \lambda xy + 2y^2 + (\lambda - 4)x + 6y – 5 = 0 \) is the equation of a circle then its radius is
(a) \( 3\sqrt{2} \)
(b) \( 2\sqrt{3} \)
(c) \( 2\sqrt{2} \)
(d) None of the options
Answer: (d) None of the options

Question. The equation \( x^2 + y^2 – 2x + 4y + 5 = 0 \) represents
(a) a point
(b) a pair of straight lines
(c) a circle of nonzero radius
(d) None of the options
Answer: (a) a point

Question. Three sides of a triangle have the equations \( L_r \equiv y - m_r x - c_r = 0; r = 1, 2, 3 \). Then \( \lambda L_2 L_3 + \mu L_3 L_1 + \nu L_1 L_2 = 0 \), where \( \lambda \neq 0, \mu \neq 0, \nu \neq 0 \), is the equation of the circumcircle of the triangle if
(a) \( \lambda(m_2 + m_3) + \mu(m_3 + m_1) + \nu(m_1 + m_2) = 0 \)
(b) \( \lambda(m_2 m_3 – 1) + \mu(m_3 m_1 – 1) + \nu(m_1 m_2 – 1) = 0 \)
(c) both (a) and (b) hold together
(d) None of the options
Answer: (c) both (a) and (b) hold together

Question. The number of integral values of \( \lambda \) for which \( x^2 + y^2 + \lambda x + (1 - \lambda)y + 5 = 0 \) is the equation of a circle whose radius cannot exceed 5, is
(a) 14
(b) 18
(c) 16
(d) None of the options
Answer: (c) 16

Question. If \( 2(x^2 + y^2) + 4\lambda x + \lambda^2 = 0 \) represents a circle of meaningful radius then the range of real values of \( \lambda \) is
(a) R
(b) \( (0, +\infty) \)
(c) \( (-\infty, 0) \)
(d) None of the options
Answer: (a) R

Question. If a circle passes through the points of intersection of the lines \( 2x – y + 1 = 0 \) and \( x + \lambda y – 3 = 0 \) with the axes of reference then the value of \( \lambda \) is
(a) 1/2
(b) 2
(c) 1
(d) -2
Answer: (d) -2

Question. The equation of the circle passing through the point (1, 1) and having two diameters along the pair lines \( x^2 – y^2 - 2x + 4y – 3 = 0 \) is
(a) \( x^2 + y^2 – 2x – 4y + 4 = 0 \)
(b) \( x^2 + y^2 + 2x + 4y – 4 = 0 \)
(c) \( x^2 + y^2 – 2x + 4y + 4 = 0 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 – 2x – 4y + 4 = 0 \)

Question. Two vertices of an equilateral triangle are (-1, 0) and (1, 0), and its third vertex lies above the x-axis. The equation of the circumcircle of the triangle is
(a) \( x^2 + y^2 = 1 \)
(b) \( \sqrt{3}(x^2 + y^2) + 2y - \sqrt{3} = 0 \)
(c) \( \sqrt{3}(x^2 + y^2) - 2y - \sqrt{3} = 0 \)
(d) None of the options
Answer: (c) \( \sqrt{3}(x^2 + y^2) - 2y - \sqrt{3} = 0 \)

Question. A triangle is formed by the lines whose combined equation is given by \( (x + y – 4)(xy – 2x – y + 2) = 0 \). The equation of its circumcircles is
(a) \( x^2 + y^2 – 5x – 3y + 8 = 0 \)
(b) \( x^2 + y^2 – 3x – 5y + 8 = 0 \)
(c) \( x^2 + y^2 - 3x – 5y – 8 = 0 \)
(d) None of the options
Answer: (b) \( x^2 + y^2 – 3x – 5y + 8 = 0 \)

Question. If the centroid of an equilateral triangle is (1, 1) and its one vertex is \( (1, 1+\sqrt{3}) \), then the equation of its circumcircle is
(a) \( x^2 + y^2 – 2x – 2y – 1 = 0 \)
(b) \( x^2 + y^2 + 2x – 2y – 3 = 0 \)
(c) \( x^2 + y^2 + 2x + 2y – 3 = 0 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 – 2x – 2y – 1 = 0 \)

Question. The equation of the circle whose one diameter is PQ, where ordinates of P, Q are the roots of the equation \( x^2 + 2x – 3 = 0 \) and the abscissa are the roots of the equation \( y^2 + 4y – 12 = 0 \), is
(a) \( x^2 + y^2 + 2x + 4y – 15 = 0 \)
(b) \( x^2 + y^2 - 4x – 2y – 15 = 0 \)
(c) \( x^2 + y^2 + 4x + 2y – 15 = 0 \)
(d) None of the options
Answer: (c) \( x^2 + y^2 + 4x + 2y – 15 = 0 \)

Question. The maximum number of points with rational coordinates on a circle whose centre is \( (\sqrt{3}, 0) \) is
(a) one
(b) two
(c) four
(d) infinite
Answer: (b) two

Question. A circle touches the y-axis at (0, 2) and has an intercept of 4 units on the positive side of the x-axis. Then the equation of the circle is
(a) \( x^2 + y^2 - 4(\sqrt{2}x + y) + 4 = 0 \)
(b) \( x^2 + y^2 – 4(x + \sqrt{2}y) + 4 = 0 \)
(c) \( x^2 + y^2 – 2(\sqrt{2}x + y) \)
(d) None of the options
Answer: (a) \( x^2 + y^2 - 4(\sqrt{2}x + y) + 4 = 0 \)

Question. \( C_1 \) is a circle of radius 1 touching the x-axis and the y-axis. \( C_2 \) is another circle of radius \( > 1 \) and touching the axes as well as the circle \( C_1 \). Then the radius of \( C_2 \) is
(a) \( 3 - 2\sqrt{2} \)
(b) \( 3 + 2\sqrt{2} \)
(c) \( 3 + 2\sqrt{3} \)
(d) None of the options
Answer: (b) \( 3 + 2\sqrt{2} \)

Question. The intercept on the line \( y = x \) by the circle \( x^2 + y^2 – 2x = 0 \) is AB. The equation of the circle with AB as a diameter is
(a) \( x^2 + y^2 + x + y = 0 \)
(b) \( x^2 + y^2 = x + y \)
(c) \( x^2 + y^2 – 3x + y = 0 \)
(d) None of the options
Answer: (b) \( x^2 + y^2 = x + y \)

Question. Two circles, each of radius 5, have a common tangent at (1, 1) whose equation is \( 3x + 4y – 7 = 0 \). Then their centres are
(a) \( (4, -5), (-2, 3) \)
(b) \( (4, -3), (-2, 5) \)
(c) \( (4, 5), (-2, -3) \)
(d) None of the options
Answer: (c) \( (4, 5), (-2, -3) \)

Question. The equation of the circumcircle of the regular hexagon whose two consecutive vertices have the coordinates (-1, 0) and (1, 0) and which lies wholly above the x-axis, is
(a) \( x^2 + y^2 – 2\sqrt{3}y – 1 = 0 \)
(b) \( x^2 + y^2 – \sqrt{3}y – 1 = 0 \)
(c) \( x^2 + y^2 – 2\sqrt{3}x – 1 = 0 \)
(d) None of the options
Answer: (a) \( x^2 + y^2 – 2\sqrt{3}y – 1 = 0 \)

Question. The equation of the incircle of the triangle formed by the axes and the line \( 4x + 3y = 6 \) is
(a) \( x^2 + y^2 – 6x – 6y + 9 = 0 \)
(b) \( 4(x^2 + y^2 – x – y) + 1 = 0 \)
(c) \( 4(x^2 + y^2 + x + y) + 1 = 0 \)
(d) None of the options
Answer: (b) \( 4(x^2 + y^2 – x – y) + 1 = 0 \)

Question. If \( p \) and \( q \) be the longest distance and the shortest distance respectively of the point (-7, 2) from any point \( (\alpha, \beta) \) on the curve whose equation is \( x^2 + y^2 – 10x – 14y – 51 = 0 \) then GM of \( p \) and \( q \) is equal to
(a) \( 2\sqrt{11} \)
(b) \( 5\sqrt{5} \)
(c) 13
(d) None of the options
Answer: (a) \( 2\sqrt{11} \)

Question. The equation of the circumcircle of an equilateral triangle is \( x^2 + y^2 + 2gx + 2fy + c = 0 \) and one vertex of the triangle is (1, 1). The equation of incircle of the triangle is
(a) \( 4(x^2 + y^2) = g^2 + f^2 \)
(b) \( 4(x^2 + y^2) + 8gx + 8fy = (1 – g)(1 + 3g) + (1 – f)(1 + 3f) \)
(c) \( 4(x^2 + y^2) + 8gx + 8fy = g^2 + f^2 \)
(d) None of the options
Answer: (b) \( 4(x^2 + y^2) + 8gx + 8fy = (1 – g)(1 + 3g) + (1 – f)(1 + 3f) \)

Question. Let \( f(x, y) = 0 \) be the equation of a circle. If \( f(0, \lambda) = 0 \) has equal roots \( \lambda = 2, 2 \) and \( f(\lambda, 0) = 0 \) has roots \( \lambda = 4/5, 5 \) then the centre of the circle is
(a) \( (2, 29/10) \)
(b) \( (29/10, 2) \)
(c) \( (-2, 29/10) \)
(d) None of the options
Answer: (b) \( (29/10, 2) \)

Question. For each \( k \in N \), let \( C_k \) denote the circle whose equation is \( x^2 + y^2 = k^2 \). On the circle \( C_k \), a particle moves \( k \) units in the anticlockwise direction. After completing its motion on \( C_k \), the particle moves to \( C_{k+1} \) in the radial direction. The motion of the particle continues in this manner. The particle starts at (1, 0). If the particle crosses the positive direction of the x-axis for the first time on the circle \( C_n \) then \( n \) is
(a) 7
(b) 6
(c) 2
(d) None of the options
Answer: (a) 7

Question. Two distinct chords drawn from the point \( (p, q) \) on the circle \( x^2 + y^2 = px + qy \), where \( pq \neq 0 \), are bisected by the x-axis. Then
(a) \( |p| = |q| \)
(b) \( p^2 = 8q^2 \)
(c) \( p^2 < 8q^2 \)
(d) \( p^2 > 8q^2 \)
Answer: (d) \( p^2 > 8q^2 \)

Question. The length of the chord of the circle \( x^2 + y^2 + 4x – 7y + 12 = 0 \) along the y-axis is
(a) 1
(b) 2
(c) 1/2
(d) None of the options
Answer: (a) 1

Question. If the line \( y – 1 = m(x – 1) \) cuts the circle \( x^2 + y^2 = 4 \) at two real points then the number of possible values of \( m \) is
(a) 1
(b) 2
(c) infinite
(d) None of the options
Answer: (c) infinite

Question. The length of the chord of the circle \( x^2 + y^2 = 9 \) passing through (3, 0) and perpendicular to the line \( y + x = 0 \) is
(a) \( 3 / \sqrt{2} \)
(b) \( 3\sqrt{2} \)
(c) \( 2\sqrt{3} \)
(d) None of the options
Answer: (b) \( 3\sqrt{2} \)

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

Question. The equation of a circle is \( x^2 + y^2= 4 \). A regular hexagon is inscribed in the circle whose one vertex is (2, 0). Then a consecutive vertex has the coordinates
(a) \( (\sqrt{3}, 1) \)
(b) \( (1, -\sqrt{3}) \)
(c) \( (\sqrt{3}, -1) \)
(d) \( (1, \sqrt{3}) \)
Answer: (b) \( (1, -\sqrt{3}) \) and (d) \( (1, \sqrt{3}) \)

Question. \( P(\sqrt{2}, \sqrt{2}) \) is a point on the circle \( x^2 + y^2 = 4 \) and Q is another point on the circle such that arc \( PQ = 1/4 \times \) circumference. The coordinates of Q are
(a) \( (-\sqrt{2}, -\sqrt{2}) \)
(b) \( (\sqrt{2}, -\sqrt{2}) \)
(c) \( (-\sqrt{2}, \sqrt{2}) \)
(d) None of the options
Answer: (b) \( (\sqrt{2}, -\sqrt{2}) \) and (c) \( (-\sqrt{2}, \sqrt{2}) \)

Question. Circles \( x^2 + y^2 = 1 \) and \( x^2 + y^2 - 8x + 11 = 0 \) cut off equal intercepts on a line through the point \( (-2, 1/2) \). The slope of the line is
(a) \( \frac{-1 + \sqrt{29}}{14} \)
(b) \( \frac{1 + \sqrt{7}}{4} \)
(c) \( \frac{-1 - \sqrt{29}}{14} \)
(d) None of the options
Answer: (a) \( \frac{-1 + \sqrt{29}}{14} \) and (c) \( \frac{-1 - \sqrt{29}}{14} \)

Question. Let \( L_1 \) be a straight line passing through the origin and \( L_2 \) be the straight line \( x + y = 1 \). If the intercepts made by the circle \( x^2 + y^2 - x + 3y = 0 \) on \( L_1 \) and \( L_2 \) are equal then which of the following equations can represent \( L_1 \)?
(a) \( x + y = 0 \)
(b) \( x – y = 0 \)
(c) \( x + 7y = 0 \)
(d) \( x – 7y = 0 \)
Answer: (b) \( x – y = 0 \) and (d) \( x – 7y = 0 \)

Question. The parametric equation of a circle is given by \( x = 3 \cos \phi + 2, y = 3 \sin \phi \). Then
(a) centre = (-2, 0)
(b) radius = 3
(c) centre = (2, 0)
(d) radius = 1
Answer: (b) radius = 3 and (c) centre = (2, 0)

Question. If A and B are two points on the circle \( x^2 + y^2 – 4x + 6y – 3 = 0 \) which are farthest and nearest respectively from the point (7, 2) then
(a) \( A = (2 - 2\sqrt{2}, -3 - 2\sqrt{2}) \)
(b) \( B = (2 + 2\sqrt{2}, -3 + 2\sqrt{2}) \)
(c) \( A = (2 + 2\sqrt{2}, -3 + 2\sqrt{2}) \)
(d) \( B = (2 - 2\sqrt{2}, -3 + 2\sqrt{2}) \)
Answer: (a) \( A = (2 - 2\sqrt{2}, -3 - 2\sqrt{2}) \) and (b) \( B = (2 + 2\sqrt{2}, -3 + 2\sqrt{2}) \)