Q 18. - Precalculus Enhanced with Graphing Utilities

Question

Write the standard form of the equation and the general form of the equation of each circle of radius $r$r and center $\left(h,k\right)$.

Graph each circle.

$r=7; (h,k)=(-5,-2)$

Step-by-Step Solution

Answer

Standard form of a circle is $(x+5{)}^2+(y+2{)}^2=7^2$

General form of the circle is $x^2+y^2+10x+4y-20=0$

Graph is as follows:

Step 1. Given information

It is given that  $r=7; (h,k)=(-5,-2)$.

Step 2. Standard form of the circle

The standard form the circle with center $\left(h,k\right)$ and radius $r$ is $(x-h{)}^2+(y-k{)}^2=r^2$

It is given that $r=7$ and $(h, k)=(-5,-2)$.

Therefore, the standard form of the circle is

$\begin{array}{rcl}(x-(-5){)}^2+(y-(-2){)}^2& =& 7^2\\ (x+5{)}^2+(y+2{)}^2& =& 7^2\end{array}$

Step 3. General form of the circle

The general form of the circle is obtained by simplifying the standard form:

$\begin{array}{rcl}\left(x^2+10x+25\right)+\left(y^2+4y+4\right)& =& 49\\ x^2+y^2+10x+4y+29& =& 49\end{array}$

Subtract $49$ from both sides

$x^2+y^2+10x+4y-20=0$

Step 4. Graph of the circle

Graph is as follows: