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 $(5,0)$. (type an ordered pair, using integers or decimals.) the radius of the circle is \\(\square\\). (type an integer or a decimal.)

Explanation:

Step1: Recall the standard circle equation

The standard form of a circle's equation 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

Given the equation \(4(x - 5)^2 + 4y^2 = 16\), we first divide every term by 4 to simplify:
\[
\frac{4(x - 5)^2}{4} + \frac{4y^2}{4} = \frac{16}{4}
\]
Simplifying each term, we get:
\[
(x - 5)^2 + y^2 = 4
\]

Step3: Identify the radius

Now that the equation is in the form \((x - h)^2 + (y - k)^2 = r^2\) (where \(h = 5\), \(k = 0\), and \(r^2 = 4\)), we solve for \(r\) by taking the square root of both sides of \(r^2 = 4\). Since \(r\) represents the radius (a non - negative quantity), we have \(r=\sqrt{4}=2\).

Answer:

2