Circles practice · Mixed difficulty

The number of common tangents to the circles one of which passes through the origin and cuts off intercepts 2 from each of the axes and the other circle has the segment joining the origin and the point $(1,1)$ as a diameter, is

If a tangent to the circle ${x^2} + {y^2} = 1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is

$f(x, y) \equiv x^2+y^2+2 a x+2 b y+c=0$ represent a circle. If $f(x, 0)=0$ has equal roots, each being 2 and $f(0, y)=0$ has 2 and 3 as its roots, then the centre of the circle is

If in a $\triangle A B C$ (whose circumcentre is origin), $a \leq \sin A$, then for any point $(x, y)$ inside the circumcircle of $\triangle A B C$

Two diameters of the circle $3{x^2} + 3{y^2} - 6x - 18y - 7 = 0$ are along the lines $3x + y = {c_1}$ and $x - 3y = {c_2}$. Then, the value of ${c_1}{c_2}$ is

The equation of the circumcircle of an equilateral triangle is $x^2+y^2+2 g x+2 f y+c=0$ and one vertex of the triangle is $(1,1)$. The equation of the incircle of the triangle, is

The line 2x -y + 1 = 0 is a tangent to the circle at the point (2, 5) and the center of the circle lies on x –2y = 4. Then the radius of the circle is

The equation to the chord of the circle ${x^2} + {y^2} = 9$ whose middle point is $\left( {1, - 2} \right)$ is

The centre of a circle passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$ is

The equation of the circle with centre $(0,2)$ and radius 2 is $x^2+y^2-m y=0$. The value of $m$ is

The equation of a circle which touches the line $y=x$ at $(1,1)$ and having $y=x-3$ as a normal, is

The equation of the circle passing through the point $(-1,2)$ and having two diameters along the pair of lines $x^2-y^2-4 x+2 y+3=0$, is

The lines $2 x-3 y=5$ and $3 x-4 y=7$ are diameters of a circle having area as 154 sq.units. Then the equation of the circle is

The number of common tangents to two circles ${x^2} + {y^2} = 4$ and ${x^2} + {y^2} - 8x + 12 = 0$ is

A line is drawn through a fixed point $P\left( {\alpha ,\beta } \right)$ to cut the circle ${x^2} + {y^2} = {r^2}$ at $A$ and $B$. Then, $PA.PB$ is equal to

The point diametrically opposite to the point $P(1,0)$ on the circle $x^2+y^2+2 x+4 y-3=0$, is

Tangents drawn from the point $(4,3)$ to the circle $x^2+y^2-2 x-4 y=0$ are inclined at an angle

If $\theta$ is the angle between the tangents from $\left( { - 1,0} \right)$ to the circle ${x^2} + {y^2} - 5x + 4y - 2 = 0$, then $\theta$ is equal to

The equation of the circle with centre $\left( {2,1} \right)$ and touching the line $3x + 4y = 5$ is

If the circles ${x^2} + {y^2} - 16x - 20y + 164 = {r^2}$ and ${(x - 4)^2} + {(y - 7)^2} = 36$ intersect at two distinct points, then

The equation of the circle whose centre is $(2,-3)$ and radius 5, is

The diameter of the circle having the pair of lines $x^2+2 x y+3 x+6 y=0$ as its normals and having the size just sufficient to contain the circle $x(x-4)+y(y-3)=0$ is

The circle passing through the point $(-1,0)$ and touching the $y$-axis at $(0,2)$ also passes through the point:

Consider $L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0$ where, $p$ is a real number, and $C: x^2+y^2+6 x-10 y+30=0$

Statement I If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$ and
Statement II If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$.

The locus of the point from which mutually perpendicular tangents can be drawn to the circle $x^2+y^2=36$ is

The inverse of the point $\left( {1,2} \right)$ with respect to the circle ${x^2} + {y^2} - 4x - 6y + 9 = 0$, is

The equations of the tangents to the circle ${x^2} + {y^2} - 6x + 4y - 12 = 0$ which are parallel to the line $4x + 3y + 5 = 0$, are

The equation $2 x^2+2 y^2+4 x+8 y+15=0$ represents

If $\left( {\alpha ,\beta } \right)$ is a point on the chord $PQ$ of the circle ${x^2} + {y^2} = 25,$ where the coordinates of $P$ and $Q$ are $\left( {3, - 4} \right)$ and $\left( {4,3} \right)$ respectively, then

Two tangents are drawn from the origin to a circle with centre at $(2,-1)$. If the equation of one of the tangents is $3 x+y=0$, the equation of the other tangent is