Chapter 1.4, Problem 18AYU

To determine

To write: The standard form and general form of the equation of the circle with radius 7 and centre $\left(-5,\mathrm{-2}\right)$

Answer to Problem 18AYU

$\text{The standard form of the equation of the circle is }\left(x\text{ + 5}\right)\text{² + }\left(y\text{ + 2}\right)\text{² = 49, and general form of the equation of the circle is }x\text{² + }y\text{² + 10}x\text{ + 4}y\text{ − 20 = 0}\text{.}$

Explanation of Solution

Given:

Radius $r\text{ = 7}$; centre $\left(h,k\right)\text{ = }\left(\text{−5, −2}\right)$

Formula used:

The standard form of an equation of a circle with radius $r$ and centre $(h,k)$ is:

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

Calculation:-

Substitute $r\text{ = 7}$ and $\left(h,k\right)=\left(\mathrm{-5},\mathrm{-2}\right)$ in the standard form of the equation.

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

$\left(x-\left(-5\right)\right)²+\left(y-\left(2\right)\right)²=7²$

$\left(x+5\right)²+\left(y+2\right)²=49$

$x²+10x+25+y²+4y+4=49$

$x²+10x+y²+4y+29-49=49-49$   Subtract 49 from both sides

$x²+y²+10x+4y\text{ − 20 }=0$

General form of the equation of the circle is $x²+y²+10x+4y\text{ − 20 }=0$.