Question

The equation of a circle is x2+y2+10x−4y−20=0 . What is the radius of the circle?

Answer 1

The equation of a circle can be the expanded form of

$\large (x-a)^2+(y-b)^2=r^2$

where $r$ is the radius of the circle, $(a,\ b)$ is the center of the circle, and $(x,\ y)$ is a point on the circle.

Here, the equation of the circle is,

$\begin{aligned}&x^2+y^2+10x-4y-20&=&\ \ 0\\ \\ \Longrightarrow\ \ &x^2+y^2+10x-4y+25+4-49&=&\ \ 0\\ \\ \Longrightarrow\ \ &x^2+y^2+10x-4y+25+4&=&\ \ 49\\ \\ \Longrightarrow\ \ &x^2+10x+25+y^2-4y+4&=&\ \ 49\\ \\ \Longrightarrow\ \ &(x+5)^2+(y-2)^2&=&\ \ 7^2\end{aligned}$

From this, we get two things:

$\begin{aligned}1.&\ \ \textsf{Center of the circle is (-5,\ 2) .}\\ \\ 2.&\ \ \textsf{Radius of the circle is \bold{7} units. }\end{aligned}$

Hence the radius is 7 units.