List of top Mathematics Questions on Circles asked in JEE Main

Let the circle \(x^2+y^2=4\) intersect the \(x\)-axis at points \(A(a,0)\) and \(B(b,0)\). Let \(P(2\cos\alpha,2\sin\alpha)\), \(0<\alpha<\frac{\pi}{2}\), and \(Q(2\cos\beta,2\sin\beta)\) be two points on the circle such that \((\alpha-\beta)=\frac{\pi}{2}\). Then the point of intersection of lines \(AQ\) and \(BP\) lies on:

Let PQ and MN be two straight lines touching the circle \(x^2+y^2-4x-6y-3=0\) at the points A and B respectively. Let O be the centre of the circle and \(\angle AOB = \pi/3\). Then the locus of the point of intersection of the lines PQ and MN is:

The absolute difference between the squares of the radii of the two circles passing through the point \( (-9, 4) \) and touching the lines \( x + y = 3 \) and \( x - y = 3 \), is equal to:

Let $ ABC $ be the triangle such that the equations of lines $ AB $ and $ AC $ are: $ 3y - x = 2 \quad \text{and} \quad x + y = 2, $ respectively, and the points $ B $ and $ C $ lie on the x-axis. If $ P $ is the orthocentre of the triangle $ ABC $, then the area of the triangle $ PBC $ is equal to:

Let $ C_1 $ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let $ C_2 $ be the circle with center $ (1, 3) $ that touches $ C_1 $ externally at the point $ (\alpha, \beta) $. If $ (\beta - \alpha)^2 = \frac{m}{n} $, and $ \gcd(m, n) = 1 $, then $ m + n $ is equal to:

Let the circle $C_{1}: x^{2}+y^{2}-2(x+y)+1=0$ and $C_{2}$ be a circle having centre at $(-1, 0)$ and radius 2. If the line of the common chord of $C_{1}$ and $C_{2}$ intersects the y-axis at the point P, then the square of the distance of P from the centre of $C_{1}$ is:

Let C be a circle with radius \( \sqrt{10} \) units and centre at the origin. Let the line \( x + y = 2 \) intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope \(-1\). Then, a distance (in units) between the chord PQ and the chord MN is

Let the centre of a circle, passing through the point \((0, 0)\), \((1, 0)\) and touching the circle \(x^2 + y^2 = 9\), be \((h, k)\). Then for all possible values of the coordinates of the centre \((h, k)\), \(4(h^2 + k^2)\) is equal to __________.

Let the line 2x + 3y – k = 0, k > 0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is x² + y² – 3x – 2y = 0 and the length of the latus rectum of the ellipse x² + 9y² = k² is m n , where m and n are coprime, then 2m + n is equal to

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :

A square is inscribed in the circle \( x^2 + y^2 - 10x - 6y + 30 = 0 \). One side of this square is parallel to \( y = x + 3 \). If \( (x_i, y_i) \) are the vertices of the square, then \( \sum (x_i^2 + y_i^2) \) is equal to:

Let the line \( L : \sqrt{2}x + y = \alpha \) pass through the point of intersection \( P \) (in the first quadrant) of the circle \( x^2 + y^2 = 3 \) and the parabola \( x^2 = 2y \). Let the line \( L \) touch two circles \( C_1 \) and \( C_2 \) of equal radius \( 2\sqrt{3} \). If the centers \( Q_1 \) and \( Q_2 \) of the circles \( C_1 \) and \( C_2 \) lie on the y-axis, then the square of the area of the triangle \( PQ_1Q_2 \) is equal to ____.

Let \( C: x^2 + y^2 = 4 \) and \( C': x^2 + y^2 - 4\lambda x + 9 = 0 \) be two circles. If the set of all values of \( \lambda \) such that the circles \( C \) and \( C' \) intersect at two distinct points is \( R = [a, b] \), then the point \( (8a + 12, 16b - 20) \) lies on the curve:

Equation of two diameters of a circle are \(2x-3y=5\) and \(3x-4y=7\).The line joining the points \((-\frac{22}{7},-4)\) and \((-\frac{1}{7},3)\) intersects the circle at only one point \(P(\alpha,\beta)\).Then \(17\beta-\alpha\) is equal to.

Let the equation $x^2 + y^2 + px + (1 - p)y + 5 = 0$ represent circles of varying radius $r \in (0, 5]$. Then the number of elements in the set $S = \{q : q = p^2 \text{ and } q \text{ is an integer}\}$ is _________.

Let B be the centre of the circle \( x^2 + y^2 - 2x + 4y + 1 = 0 \). Let the tangents at two points P and Q on the circle intersect at the point \( A(3, 1) \). Then \( 8 \cdot \frac{\text{area } \Delta APQ}{\text{area } \Delta BPQ} \) is equal to _________.

Let the lines \( (2 - i)z = (2 + i)\bar{z} \) and \( (2 + i)z + (i - 2)\bar{z} - 4i = 0 \), (here \( i^2 = -1 \)) be normal to a circle \( C \). If the line \( iz + \bar{z} + 1 + i = 0 \) is tangent to this circle \( C \), then its radius is :

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept $6\sqrt{5}$ on the x-axis. Then the radius of the circle C is equal to :