Chapter 1.5, Problem 28AYU

To determine

To find:

1. The centre $\left(h,k\right)$ and radius of the circle $x²+y²+4x+2y-20=0$

2. The graph of the circle

3.Intercepts of the circle, if any

Answer to Problem 28AYU

$\text{Centre }\left(-2,-1\right);\text{ radius }=5$

$x\text{-intercepts }=\sqrt{24}-2\text{ and }-\sqrt{24}–2;y\text{-intercepts }=\sqrt{21}-1\text{ and }-\sqrt{21}-1$

Explanation of Solution

Given:

Equation of the circle $x²+y²+4x+2y-20=0$

Calculation:

$x²+y²+4x+2y-20=0$

$\left(x²+4x\right)+\left(y²+2y\right)-20=0$

$(x²+4x)+(y²+2y)-20+20=0+20$   Add 20 to both sides

$\left(x²+4x+4\right)+\left(y²+2y+1\right)=20+5$   Complete the square of each expression in parenthesis

$\left(x+2\right)²+\left(y+1\right)²=\left(5\right)²$

Compare this equation with the equation $\left(x-h\right)²+\left(y-k\right)²=r²$.

The comparison yields the information about the circle. We see that $h=-2,k=-1$ and $r=5$.

The circle has centre $\left(\mathrm{-2},\mathrm{-1}\right)$ and a radius 5 units. To graph the circle, first plot the centre $\left(\mathrm{-2},\mathrm{-1}\right)$. Since the radius is 5 units, locate four points on the circle by plotting 5 units to the left, to the right, up and down from the centre. These four points can be used to sketch the graph.

To find the $x\text{-intercepts}$, if any, let $y\text{= }0$ and solve for $x$.

$\left(x+2\right)²+\left(0+1\right)²=\left(5\right)²$

$\left(x+2\right)²+\left(0+1\right)²=\left(5\right)²$

$\left(x+2\right)²+1=25$

$\left(x+2\right)²+1-1=25-1$   Subtract 1 from both sides

$\left(x+2\right)²=24$

$x+2=\pm \sqrt{24}$

$\text{If},x+2=\sqrt{24}$

$x+2-2=\sqrt{24}-2$   Subtract 2 from both sides

$x=\sqrt{24}-2$

$\text{If},x+2=-\sqrt{24}$

$x+2-2=-\sqrt{24}-2$   Subtract 2 from both sides

$x=-\sqrt{24}-2$

$x\text{-intercepts}$ are $\sqrt{24}-2$ and $-\sqrt{24}–2$.

To find $y\text{-intercepts}$, if any, let $x=0$ and solve for $y$.

$\left(0+2\right)²+\left(y+1\right)²=25$

$4+\left(y+1\right)²=25$

$\left(y+1\right)²+4-4=25-4$   Subtract 4 from both sides

$\left(y+1\right)²=21$

$y+1=\pm \sqrt{21}$

$\text{If }y+1=\sqrt{21}$

$y+1-1= \sqrt{21}-1$

$y= \sqrt{21}-1$

$\text{If }y+1=-\sqrt{21}$

$y+1-1= -\sqrt{21}-1$

$y= -\sqrt{21}-1$

$y\text{-intercepts}$ are $\sqrt{21}-1$ and $-\sqrt{21}–1$.