Question

a circle has the equation $4(x - 5)^2 + 4y^2 = 16$. (a) find the center $(h,k)$ and radius $r$ of the circle. (b) graph the circle. (c) find the intercepts, if any, of the graph. (a) the center of the circle is \boxed{}. (type an ordered pair, using integers or decimals.)

Explanation:

Step1: Recall the standard form of a circle's equation

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

Step2: Rewrite the given equation in standard form

We are given the equation \(4(x - 5)^2+4y^2 = 16\). First, divide the entire equation by 4 to simplify:

$$ \frac{4(x - 5)^2}{4}+\frac{4y^2}{4}=\frac{16}{4} $$

This simplifies to \((x - 5)^2+y^2 = 4\). We can rewrite \(y^2\) as \((y - 0)^2\) and \(4\) as \(2^2\), so the equation becomes \((x - 5)^2+(y - 0)^2=2^2\).

Answer:

\((5,0)\)