List of top Mathematics Questions on Circle

The number of common tangents to the circles \[ x^2+y^2-4x-6y-12=0,\quad x^2+y^2+6x+18y+26=0 \] is:
Let \( x-y=0 \) and \( x+y=1 \) be two perpendicular diameters of a circle of radius \( R \). The circle will pass through the origin if \( R \) equals:
The maximum number of common normals of \( y^2 = 4ax \) and \( x^2 = 4by \) is:
Find the area of a circle with radius 7 cm.
Find the equation of the circle which passes through the points $ (1,2) $, $ (4,3) $ and has its center on the line $ x + y = 5 $.
If the circles \( (x+1)^2 + (y+2)^2 = r^2 \) and \( x^2 + y^2 - 4x - 4y + 4 = 0 \) intersect at exactly two distinct points, then
If one of the diameters of the circle \(x^2+y^2-10x+4y+13=0\) is a chord of another circle and whose center is the point of intersection of the lines \(2x + 3y = 12\) and \(3x - 2y = 5 \) then the radius of the circle is
With origin as a focus and x = 4 as the corresponding directrix, a family of ellipses are drawn. Then the locus of an end of the minor axis is:
Chords AB CD of a circle intersect at right angle at the point P. If the lengths of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is:
If two circles which pass through the points \( (0, a) \) and \( (0, -a) \) and touch the line \( y = mx + c \) cut orthogonally, then:

Four distinct points \( (2k, 3k), (1, 0), (0, 1) \) and \( (0, 0) \) lie on a circle for \( k \) equal to:

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3(\alpha+\beta+\gamma)$ is equal to
Let $y=x+2,4 y=3 x+6$ and $3 y=4 x+1$ be three tangent lines to the circle $(x-h)^2+(y- k )^2= r ^2$ Then $h + k$ is equal to :
The points of intersection of the line $a x+b y=0,(a \neq b)$ and the circle $x^2+y^2-2 x=0$ are $A (\alpha, 0)$ and $B (1, \beta)$ The image of the circle with $AB$ as a diameter in the line $x+y+2=0$ is :
The points of intersection of the line $a x+b y=0,(a \neq b)$ and the circle $x^2+y^2-2 x=0$ are $A (\alpha, 0)$ and $B (1, \beta)$ The image of the circle with $AB$ as a diameter in the line $x+y+2=0$ is :
The locus of the mid points of the chords of the circle $C_1:(x-4)^2+(y-5)^2=4$ which subtend an angle $\theta_i$ at the centre of the circle $C_1$, is a circle of radius $r_i$ If $\theta_1=\frac{\pi}{3}, \theta_3=\frac{2 \pi}{3}$ and $r_1^2=r_2^2+r_3^2$, then $\theta_2$ is equal to
The distance of the point $(6,-2 \sqrt{2})$ from the common tangent $y=m x+c, m>$, of the curves $x=2 y^2$ and $x=1+y^2$ is :
A circle passing through the point P\(( \alpha , \beta )\) in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then value of \(( \alpha \beta )\)  is ___
Two circles having radius \(r_1\) and \(r_2\) touch both the coordinate axes. Line \(x+y=2\) makes intercept 2 on both the circles. The value of \(r_{1}^{2} + r_{2}^{2} - r_{1}r_{2}\) is:
Let A be the point (1, 2) and B be any point on the curve \( x^2 + y^2 = 16 \). If the centre of the locus of the point P, which divides the line segment AB in the ratio 3:2 is the point C (\( \alpha, \beta \)), then the length of the line segment AC is:

Let O be the origin and OP and OQ be the tangents to the circle \( x^2 + y^2 - 6x + 4y + 8 = 0 \) at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point \( \left( \alpha, \frac{1}{2} \right) \), then a value of \( \alpha \) is.

Let C be the circle in the complex plane with centre z0 =\(\frac{1}{2}\) (1+3i) and radius r = 1. Let z1 = 1+ i and the complex number z2 be outside the circle C such that |z1 – z0| |z2 – z0| = 1. If z0, z1 and z2 are collinear, then the smaller value of |z2|2 is equal to
For α, β, z ∈ C and λ > 1, if \(\sqrt{λ - 1}\) is the radius of the circle |z - α|2 + |z - β|2 = 2λ, then |α - β| is equal to _____.
Let the centre of a circle C be (α, β) and its radius r < 8. Let 3x + 4y = 24 and 3x – 4y = 32 be two tangents and 4x + 3y =1 be a normal to C. Then (α – β + r) is equal to
Let C be the circle in the complex plane with centre z0 = \(\frac{1}{2}\) (1+3i) and radius r = 1. Let z1 = 1+ i and the complex number z2 be outside the circle C such that |z1 – z0| |z2 – z0| = 1. If z0, z1 and z2 are collinear, then the smaller value of |z2|2 is equal to