Question

If $x+2y=7$ and $2x+y=8$ are the equations of the lines of two diameters of a circle, find the radius of the circle if the point $(0,-2)$ lies on the circle.

Answer 1

Let $C_1$ be a circle with center $(a,b)$.

Now given that $x+2y=7$ and $2x+y=8$ two equations of diameters of $C_1$. Since both are diameters then center will satisfy, therefore

$a+2b=7-------------------\left(1\right)$

$2a+b=8-------------------\left(2\right)$

Now, solving equation (1) and (2)

So, $a=3,b=2$ Center of $C_1=(3,2)$.

Since, $(0,-2)$ lies on circle $C_1$ then distance between $(0,-2)and(3,2)$ is equal to radius $r$(say) of circle.

Now,

$r=\sqrt{(3-0{)}^2+(2+2{)}^2}$

$r=\sqrt{9+16}$

$r=\sqrt{25}$

$r=5$

So, radius of circle is $5$.