Circles practice · Mixed difficulty

If $S_1: x^2+y^2=4$ and $S_2: x^2+y^2-2 a x-2 b y+2=0$ touches each other, then 4 times of radius of the circle $S_2$ is:

The two circles ${x^2} + {y^2} - 2x - 2y + 1 = 0$ and ${x^2} + {y^2} - 4x - 6y - 8 = 0$ are such that

Tangents at two points $A$ and $B$ on the circle $x^2+$ $y^2-6 x-4 y-11=0$ intersect at the point $(1,8)$. Equation of the chord $A B$ is

Find the length of the tangent drawn from the point $(5,1)$ to the circle $x^2+y^2+6 x-4 y-3=0$.

Combined equation of tangents drawn to $x^2+y^2=1$ from $(-1,-1)$, is

The locus of the centre of the circle for which one end of a diameter is $\left( {1,1} \right)$ while the other end is on the line $x + y = 3$, is

A circle having centre $(0, k)$ with $k>6$ is tangent to the line pair $x^2-y^2=0$ and $y=6$. the radius of the circle is

If the incentre of an equilateral triangle is $(1,1)$ and the equation of its one side is $3 x+4 y+3=0$, then the equation of the circumcircle of this triangle is :

If the lines $2 x-3 y-5=0$ and $3 x-4 y=7$ are diameters of a circle of area 154 square units, then the equation of the circle is

The equation of the tangents to the circle ${x^2} + {y^2} = 4$, which are parallel to $x + 2y + 3 = 0$, are

The length of the longest ray drawn from the point $(4,3)$ to the circle $x^2+y^2+16 x+18 y+1=0$ is equal to

The coordinates of the centre of the smallest circle passing through the origin and having $y = x + 1$ as a diameter are

If the square of the length of the tangent drawn from any point on the circle $x^2+y^2+70 x-45 y+6789=0$ to the circle $x^2+y^2+70 x-45 y+9876=0$ is $K$ then find $K-3080.$

The radius of the circle ${r^2} - 2\sqrt 2 r\left( {\cos \theta + \sin \theta } \right) - 5 = 0,$ is

Find the equation to the circle touching the $y$-axis at a distance -3 from the origin and intercepting a length 8 on the $x$-axis.

If $(2,4)$ is a point interior to the circle $x^2+y^2-6 x-10 y+\lambda=0$ and the circle does not cut the axes at any point, then

In the two circles ${\left( {x + 7} \right)^2} + {\left( {y - 3} \right)^2} = 36$ and ${\left( {x - 5} \right)^2} + {\left( {y + 2} \right)^2} = 49$ touch each other externally, then the point of contact is

$\alpha, \beta$ and $\gamma$ are parametric angles of three points $P, Q$ and $R$ respectively, on the circle $x^2+y^2=1$ and $A$ is the point $(-1,0)$. If the lengths of the chords $A P, A Q$ and $A R$ are in $G P$, then $\cos \left(\frac{\alpha}{2}\right), \cos \left(\frac{\beta}{2}\right)$ and $\cos \left(\frac{\gamma}{2}\right)$ are in

If $P\left( { - 1, - 3} \right)$ is a centre of similitude for the circles ${x^2} + {y^2} = 1$ and ${x^2} + {y^2} - 2x - 6y + 6 = 0,$ then the length of the common tangent through $P$ to the circles is

If the points $\left( {2,0} \right),\left( {0,1} \right),\left( {4,5} \right)$ and $\left( {0,c} \right)$ are concyclic, then the value of $c$ is

The equation of the tangent to circle $5{x^2} + 5{y^2} = 1$, parallel to line $3x + 4y = 1$ are

Let $P Q$ and $R S$ be tangents at the extremities of diameter $P R$ of a circle of radius $r$. If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals

The radius of any circle touching the lines $3x - 4y + 5 = 0$ and $6x - 8y - 9 = 0$ is

If the centroid of an equilateral triangle is $(2,-2)$ and its one vertex is $(-1,1)$, then the equation of its circumcircle is

Let $S$ be the circle in the $x y$-plane defined by the equation $x^2+y^2=4$. Let $P$ be a point on the circle with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at $M$ and $N$. Then the mid-point of the line segment MN must lie on the curve

Radius of bigger circle touching the circle $x^2+y^2-4 x-4 y+4=0$ and both the coordinate axes, is

The number of points with integral coordinates $\left( {2a,a - 1} \right)$ that fall in the interior of the larger segment of the circle ${x^2} + {y^2} = 25$ cut off by the parabola ${x^2} + 4y = 0,$ is

The angle between the tangents drawn at the points $\left( {5,12} \right)$ and $\left( {12, - 5} \right)$ to the circle ${x^2} + {y^2} = 169$ is

The values of $\lambda$ so that the line $3x - 4y = \lambda$ touches ${x^2} + {y^2} - 4x - 8y - 5 = 0$ are

Let the tangent to the circle $C_1: x^2+y^2=2$ at the point $M(-1,1)$ intersect the circle $C_2:(x-3)^2+(y-$ $2)^2=5$, at two distinct points $A$ and $B$. If the tangents to $C_2$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $A N B$ is equal to