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.

$r=5;(h,k)=(4,-3)$

Short answer

Standard form of the circle is $(x-4{)}^2+(y+3{)}^2=5^2$

General form of the circle is$x^2+y^2-8x+6y=0$

Graph is as follows:

Step-by-step solution

Step 1. Given information

It is given that$r=5;(h,k)=(4,-3)$

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 $r=5$and $\left(h,k\right)=\left(4,-3\right)$.

Therefore, the standard form of the circle is $(x-4{)}^2+(y-(-3){)}^2=5^2$

$(x-4{)}^2+(y+3{)}^2=5^2$

Step 3. General form of the circle

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

$\begin{array}{rcl}\left(x^2-8x+16\right)+\left(y^2+6y+9\right)& =& 25\\ x^2+y^2-8x+6y+25& =& 25\end{array}$

Subtract $25$from both sides

$x^2+y^2-8x+6y=0$

Step 4. Graph of the circle

Graph is as follows: