Chapter 1, Problem 105RE

To determine

To calculate:The equation for the given circle with center $\left(-5,-1\right)$ and which also passes through the origin.

Answer to Problem 105RE

The equation for the given circle with center $\left(-5,-1\right)$ and passing through origin is:

  $x^2+y^2+10x+2y=0$

Explanation of Solution

Given information:

A circle with center $\left(h,k\right)=\left(-5,-1\right)$ and it passes through origin.

Formula used:

For a given circle with center $\left(h,k\right)$ and radius $r$ the equation is given as:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$

This is referred to as the Standard form for the equation of a given circle.

But, when an extra condition that is the circle passes through origin is imposed on the standard equation then with the help of distance formula to get $D$ between two points say $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given as:

  $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Then we get $r^2=h^2+k^2$

Putting this in standard equation we get:

The equation for the given circle with center $\left(h,k\right)$ and which also passes through the origin is:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=h^2+k^2$

Calculation:

Consider the given circle with center $\left(h,k\right)=\left(-5,-1\right)$ and passing through origin.

Recall that the equation for the given circle with center $\left(h,k\right)$ and which also passes through the origin is:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=h^2+k^2$

Therefore, replacing the values we get:

  ${\left(x-\left(-5\right)\right)}^2+{\left(y-\left(-1\right)\right)}^2={\left(-5\right)}^2+{\left(-1\right)}^2$ $\left(1\right)$

Now using the following formulae:

  $\begin{array}{l}{\left(a+b\right)}^2=a^2+2ab+b^2\\ {\left(a-b\right)}^2=a^2-2ab+b^2\end{array}$

Then, $\left(1\right)$ reduces to:

  $\begin{array}{l}{x}^2+25+10x+y^2+1+2y=25+1\\ x^2+y^2+10x+2y+26=26\\ x^2+y^2+10x+2y+26-26=0\\ x^2+y^2+10x+2y=0\end{array}$

Hence, the equation for the given circle with center $\left(-5,-1\right)$ and passing through origin is:

  $x^2+y^2+10x+2y=0$