List of top Mathematics Questions on Circles

Prove that tangent segments drawn from an external point to a circle are congruent.

\(\square\)ABCD is a rectangle. Taking AD as a diameter, a semicircle AXD is drawn which intersects the diagonal BD at X. If AB = 12 cm, AD = 9 cm, then find the values of BD and BX.

In cyclic quadrilateral ABCD m\(\angle\)A = 100\(^\circ\), then find m\(\angle\)C.

\(\angle\)ACB is inscribed angle in a circle with centre O. If \(\angle\)ACB = 65\(^{\circ}\), then what is measure of its intercepted arc AXB ?

The area of the circle whose circumference is equal to the perimeter of a square of side 11 cm is:

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.

If a hexagon ABCDEF circumscribes a circle, prove that AB + CD + EF = BC + DE + FA.

The minute hand of a clock is 21 cm long. The area swept by it in 10 minutes is :