Family of Circles

Tangents $TP$ and $TQ$ are drawn from a point $T$ to circle $x^2+y^2=4$. If point $T$ lies on $3x+4y-12=0$, then the area (in sq. units) of figure formed by locus of centre of circumcircle of $∆TPQ$ and co-ordinate axes is

Find the equation of a circle which passes through the point $(1, 2)$ and the points of intersection of the circles $x^2+y^2-8x-6y+21=0$ and $x^2+y^2-2x-15=0.$

The circle described on the chord $x\mathrm{ }\mathrm{cos\alpha }+y\mathrm{ }\mathrm{sin\alpha }-p=0$ of the circle $x^2+y^2=a^2$ as diameter passes through the origin if

A circle $c$ passing through the point $\left(6,2+2\sqrt{3}\right)$ touches the circle $x^2+y^2-2x-4y-4=0$ externally at the point $\left(4,2\right)$, then diameter of circle $c$ is

If $x+y=3$ is the equation of the chord $AB$ of the circle $x^2+y^2-2x+4y-8=0$, then the equation of the circle having $AB$ as diameter is

The straight line $x \cos \alpha +y \sin \alpha =p$ is intercepted by the circle $x^2+y^2=a^2$ to form a chord of the circle. Taking this chord as a diameter, a circle is drawn. Prove that the equation of the circle drawn is $x^2+y^2-2p\left(x \cos \alpha +y \sin \alpha \right)=a^2-2p^2$.

The circle $x^2+y^2+2x-4y-11=0$ and the line $x-y+1=0$ intersect at $A$ and $B$. Find the equation of the circle, on $AB$ as diameter.  

Center of the circle which passes through the point $\left(-2,2\right)$ and touches the circle $x^2+y^2-6x+8y=0$ at the point $\left(0,0\right)$ will be

The radius of the circle touching the line $\mathrm{x}+\mathrm{y}=4$ at $\left(\mathrm{1,3}\right)$ and intersecting ${\mathrm{x}}^2+{\mathrm{y}}^2=4$ orthogonally is

If the diameter of first circle is the common chord for other circles having equation $x^2+y^2+2x+3y+1=0$ and $x^2+y^2+4x+3y+2=0$, then find the equation of first circle

If the circle $x^2+y^2+6x+8y-5=0$ is concentric with the circle $x^2+y^2+2gx+2fy+c=0$ on which the point $(-3, 2)$ lies, then the value of $c$ will be:

The coordinates of the two fixed points through which the circle $x^2+y^2-2x+\lambda x-1=0$ passes are

Cotyledons are also called-

Consider a family of circles passing through two fixed points $A\left(3,7\right)$ and $B\left(6,5\right)$. The chord in which the circle $x^2+y^2-4x-6y-3=0$ cuts each member of family of circles passes through a fixed point $\left(a,b\right)$. Then the value of $a+3b$ is

The $P$ be the family of parabolas $y=x^2+px+q \left(q \ne 0\right)$ whose graphs cut the axes in three points. The family of circles through these three points have a common point

The equation of the circle whose diameter is the common chord of the circles ${\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{x}+3\mathrm{y}+2=0$ and ${\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{x}-3\mathrm{y}-\mathrm{ }4=0$ is-

A triangle is formed by the lines whose combined equation given by $\left(x+y- 4\right)\left(xy- 2x- y+2\right)= 0$. The equation of circumcircle is -

The circle ${\mathrm{x}}^2+{\mathrm{y}}^2=1$ is completely contained in the circle ${\mathrm{x}}^2+{\mathrm{y}}^2+4\mathrm{x}+3\mathrm{y}+\mathrm{k}=0$ , if

The equation of circle passing through the points of intersection of circles $x^2+y^2-6x+8=0$ and $x^2+y^2=6$ and point $\left(\mathrm{1,1}\right)$ is -

If a circle passes through the points where the lines $3kx-2y-1=0$ and $4x-3y+2=0$ meet the coordinate axes then positive value of $k$ is