Q 16. - 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 $r$ and center $\left(h,k\right)$.

Graph each circle.

$r=4; (h,k)=(2,-3)$

Step-by-Step Solution

Answer

Standard form of the circle is $(x-2{)}^2+(y+3{)}^2=4^2$

General form of the circle is $x^2+y^2-4x+6y-3=0$

Graph is as follows:

Step 1. Given information

It is given that $r=4; (h,k)=(2,-3)$

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 $r=4$ and $\left(h,k\right)=\left(2,-3\right)$.

Therefore, the standard form of the circle is

$\begin{array}{rcl}(x-2{)}^2+(y-(-3){)}^2& =& 4^2\\ (x-2{)}^2+(y+3{)}^2& =& 4^2\end{array}$

Step 3. General form of the circle

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

$\begin{array}{rcl}\left(x^2-4x+4\right)+\left(y^2+6y+9\right)& =& 16\\ x^2+y^2-4x+6y+13& =& 16\end{array}$

Subtract $16$ from both sides

$x^2+y^2-4x+6y-3=0$

Step 4. Graph of the circle

Graph of the circle is as follows: