Question

2.2
score: 12/13 answered: 12/13
question 13
for the circle having the given equation, find each coordinate of the center exactly. also find radius either exactly with \sqrt()\ or by rounding to 3 decimal places.
$x^{2}+y^{2}+4x + 8y-25 = 0$
center: (
radius:
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Explanation:

Step1: Rewrite the equation in standard form

Complete the square for \(x\) and \(y\) terms.
\[

$$\begin{align*} x^{2}+y^{2}+4x + 8y-25&=0\\ x^{2}+4x+y^{2}+8y&=25\\ x^{2}+4x + 4+y^{2}+8y+16&=25 + 4+16 \end{align*}$$

\]

Step2: Factor the left - hand side

\((x + 2)^{2}+(y + 4)^{2}=45\)

The standard form of a circle equation is \((x - a)^{2}+(y - b)^{2}=r^{2}\), where \((a,b)\) is the center of the circle and \(r\) is the radius.

Answer:

Center: \((-2,-4)\)
Radius: \(\sqrt{45}\approx6.708\)