Exercise-1

Find the equation of a circle which touches the line $2x-y=4$ at the point $\left(1,-2\right)$ and whose radius $=5$

Find the equation of the circle circumscribing the triangle formed by the lines $x+y=6, 2x+y=4$ and $x+2y=5$.

Prove that the two circles $x^2+y^2+2by+c^2=0\text{ and }{x}^2+y^2+2ax+c^2=0$ will touch each other if $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$.

The locus of mid-points of chord of the circle $x^2+y^2=4$ which subtends a right angle at the origin is

The locus of the midpoints of the chords of the circle $x^2+y^2=4$ such that the segment intercepted by the chord on the curve $x^2=2\left(x+y\right)$ subtends a right angle at the origin, is

The equation to the circle cutting orthogonally the three circles $x^2+y^2-2x+3y-7=0, x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$, is

Two circles are drawn through the point $\left(a,5a\right)$ and $\left(4a,a\right)$ to touch the axis of ${}^{\text{'}}y^{\text{'}}.$ They intersect at an angle of $\theta$ then $\tan \theta$ is

A parabola $y=a{x}^2+bx+c$ crosses the $x$-axis at $\left(\alpha ,0\right) \left(\beta ,0\right)$ both to the right of the origin. A circle also passes through these two points. Find the length of a tangent from the origin to the circle.