Find the equation of a circle with radius 5 units and touching the circle $x^2+y^2-2 x-4 y-20=0$ at the…
Find the equation of a circle with radius 5 units and touching the circle $x^2+y^2-2 x-4 y-20=0$ at the point $(5,5)$.
$x^2+y^2-18 x-16 y+120=0$
$x^2+y^2+18 x+16 y-120=0$
$x^2+y^2-18 x+16 y-120=0$
$x^2+y^2+18 x+16 y+120=0$
Solution
Equation of given circle
$
\begin{array}{rlrl}
x^2+y^2-2 x-4 y-20 & =0 \\
\Rightarrow & & (x-1)^2+(y-2)^2 & =25
\end{array}
$
Having centre $C_1(1,2)$ and radius 5 units.
$\because$ Another circle touches the circle (i) at point $(5,5)$ having radius 5 units, so centre ' $C_2$ ' of the second circle is collinear with $C_1$ and point $(5,5)$ is the mid-point of $C_1 C_2$, so $C_2(9,8)$.
So, equation of the second circle is
$
\begin{aligned}
(x-9)^2+(y-8)^2 & =25 \\
\Rightarrow \quad x^2+y^2-18 x-16 y+120 & =0
\end{aligned}
$