Family of Circles

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

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

Show that the equation of the circle described on the chord $x \cos \alpha +y \sin \alpha =p$ of the circle $x^2+y^2=a^2$ as diameter is $x^2+y^2-a^2-2p(x \cos \alpha +y \sin \alpha -p)=0$.

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 $x^2+y^2-2p\left(x \cos \alpha +y \sin \alpha \right)=a^2-2p^2$.

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

Find the equation of the circle whose center lies on $3x+4y=7$ and which passes through the points $(1, -2)$ and $(4, -3)$.

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

Consider a family of circles passing through two fixed points $A\left(3,7\right)$ and $B\left(6,5\right)$. If the common chords of the circle $x^2+y^2-4x-6y-3=0$ and the members of the family of circles pass through a fixed point $\left(a,b\right)$, then

The equation to the circle which touches the axis of $y$ at the origin and passes through $\left(3, 4\right)$ is

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 equation of the circles passing through $\left(1, 2\right)$ and the points of intersection of the circles $x^2+y^2-8x-6y+21=0$ and $x^2+y^2-2x-15=0$ is

If $x-y+1=0$ meets the circle $x^2+y^2+y-1=0$ at A and B then the equation of the circle with AB as diameter is

Find the equation of the circle through the point $\left(-2, 4\right)$ and through the points of intersection of the circle $x^2+y^2+2x-4y-4=0$ and the line $3x+2y-5=0$

Equation of the circle, whose diameter is the chord $x+y=1$ of the circles $x^2+y^2=4$ , is -

Equation of smallest circle passing through 'points of intersection of line $x+y=1$ & circle $x^2+y^2=9$

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-