List of top Mathematics Questions on Circles

The radical axis of two orthogonal circles is \(x+1=0\). If one of those circles is \(x^2+y^2=4\), then the equation of the other circle is:

Equation of the circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length \(3\) is:

If the radical axis of the circles \(x^2+y^2+2gx+2fy+c=0\) and \(2x^2+2y^2+3x+8y+2c=0\) touches the circle \(x^2+y^2+2x+2y+1=0\), then:

The tangent to the circle \(C_1:x^2+y^2-2x-1=0\) at the point \((2,1)\) cuts off a chord of length \(4\) units from a circle \(C_2\) whose centre is \((3,-2)\). The radius of circle \(C_2\) is:

If the chord \(y=mx+1\) of the circle \(x^2+y^2=1\) subtends an angle \(45^\circ\) at the major segment of the circle, then the value of \(m\) is:

\(T_1,T_2\) are points of contact of a transverse common tangent drawn to circles \[ x^2+y^2+4x-10y+4=0 \] and \[ x^2+y^2-6x+8y+9=0 \] If \(T_1T_2\) is horizontal line, midpoint of segment \(T_1T_2\) is

The external centre of similitude for circles \[ x^2+y^2+10x-16y-11=0 \] and \[ x^2+y^2-2x+4y-4=0 \] is

Let \(L_{1}\equiv3x+4y-1=0,\; L_{2}\equiv8x-6y+1=0,\; L_{3}\equiv12x+9y-1=0\) be three tangents drawn to the circle \[ x^2+y^2+2gx+2fy+c=0 \] and \(L_1>0,\;L_2>0,\;L_3>0\) at the centre \((-g,-f)\). Then \(g+2f=\)

A circle \(S\) passing through origin cuts another circle \[ x^2+y^2-6x+8y+16=0 \] orthogonally and makes a chord of maximum length on line \[ x-y-2=0 \] then one diameter of circle \(S\) is

If \[ l_1x+m_1y+n_1=0 \] and \[ l_2x+m_2y+n_2=0 \] are tangents drawn from point \((2,-1)\) to circle \[ x^2+y^2=4 \] then \(n_1+n_2=\)

If circle \[ x^2+y^2+2x+4y+k=0 \] lies totally inside third quadrant and point \[ \left(-\frac12,-\frac12\right) \] lies outside circle, then set of all real values of \(k\) is

The centre of the circle which intersects the circles \[ x^2+y^2-8x+10y+5=0 \] and \[ x^2+y^2-2x+2y+1=0 \] orthogonally is:

The locus of the point which forms a right-angled triangle with the fixed points \((2,3)\) and \((5,1)\) is

A circle C cuts the circles \(x^{2}+y^{2}-4x+6y+4=0\) and \(x^{2}+y^{2}+6x-4y+9=0\) orthogonally. If origin lies on this circle C, then the radius of the circle C is

Two circles, each of radius 5, touch at $(1,2)$. If the common tangent at the point of contact is $4x+3y=10$, then the equation of one of the circles is:

If a circle \(S\) passes through \((a, b)\) and cuts the circle \(x^2 + y^2 = 4\) orthogonally, then find the locus of the center of \(S\).

The sum of the squares of the lengths of the chords intercepted on \(x^2+y^2=16\) by the lines \(x+y=n, n \in \mathbb{N}\) is:

Find the interval of \(\lambda\) for which exactly two common tangents can be drawn to \(x^2+y^2-4x-4y+6=0\) and \(x^2+y^2-10x-10y+\lambda=0\).

A variable circle passes through the fixed point \(A(p,q)\) and touches the X-axis. The locus of the other end of the diameter through A is

Let \( \theta \) be the angle between the circles \( x^{2}+y^{2}-4x+2fy-f=0 \) and \( x^{2}+y^{2}+2fx-4y-f=0 \). If \( \cos\theta=\frac{9}{16} \) and \( f\in\mathbb{Z} \), then the distance between the centres of these circles is

The line \( 5x-12y-4=0 \) cuts the circle \( x^{2}+y^{2}-2x+2y+c=0 \) at two points A, B. If \( AB=2\sqrt{3} \), then the length of the tangent drawn from the point (2, 1) to the given circle is

The sum of the slopes of the common tangents drawn to the circles \( x^{2}+y^{2}+4x-2y-11=0 \) and \( x^{2}+y^{2}-2x+6y+6=0 \) is

The perpendicular distance from origin to the tangent drawn at the point \( P(\frac{\pi}{4}) \) to the circle \( x^{2}+y^{2}-4x-4y+6=0 \) is

If \( 2x+y-2=0 \) and \( 6x-4y+1=0 \) are two normals of a circle S and the length of the perpendicular drawn from (2, 3) to the line \( 3x+4y-3=0 \) is the radius of S, then the interior point of the circle S among the following options is

The locus of the centre of a circle of radius 2 which rolls on the outside of the circle \( x^{2}+y^{2}+3x-6y-9=0 \) along its circumference is