Q 13. - 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.

$r=2; (h,k)=(0,2)$

Step-by-Step Solution

Answer

Standard form of a circle is $(x-0{)}^2+(y-2{)}^2=2^2$

General form of a circle is $x^2+y^2-4y=0$

Graph is as follows:

Step 1. Given information

It is given that $r=2; (h,k)=(0,2)$

Step 2. Standard form of the circle

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

It is given that $r=2$ and $\left(h,k\right)=\left(0,2\right)$.

Therefore, the standard form of the circle is $(x-0{)}^2+(y-2{)}^2=2^2$

Step 3. General form of the circle

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

$x^2+\left(y^2-4y+4\right)=4$

Subtract $4$ from both sides

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

Step 4. Graph of the circle

Graph is as follows: