Question

If the line $2x-y+1=0$ touches the circle at point $(2,5)$ and the centre of the circle lies on the line $x+y-9=0$. Find the equations of the circle.

Answer 1

$2x-y+1=0$ is the Equation of tangent

equation of normal

$y-y_1=m(x-x_1)$

$y-5=\frac{-1}{2}(x-2)$

$y-5=\frac{-x}{2}+1$

$2y-10+x-1=0$

$2y+x-11=0$

(equation of normal )

$\Rightarrow x+y=9$

$2y+x=11$

$-$ $-$ $-$

____________

$-y=-2$

$y=2$

so $x=7$

center $=(7,2)$

$r=\sqrt{(7-2{)}^2+(2-5{)}^2}=\sqrt{25+9}$

$r=\sqrt{34}$

Equation of circle

$(x-7{)}^2+(y-2{)}^2=34$