Question

in the xy - plane, circle m is the graph of the equation $(x - 5)^2+(y - 4)^2 = 9$. circle p has the same center as circle m but has a radius that is twice the radius of circle m. which equation represents circle p?
a $(x - 5)^2+(y - 4)^2 = 9$
b $(x - 5)^2+(y - 4)^2 = 18$
c $(x - 10)^2+(y - 8)^2 = 9$
d $(x - 10)^2+(y - 8)^2 = 18$

Explanation:

Step1: Find the radius of circle M

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.
For circle \(M\): \((x - 5)^2+(y - 4)^2 = 9\), so \(r^2=9\), then \(r = 3\).

Step2: Calculate the radius of circle P

Since the radius of circle \(P\) is twice the radius of circle \(M\), \(r_{P}=2\times3 = 6\).
Then \(r_{P}^2=36\).

Step3: Determine the equation of circle P

Circle \(P\) has the same center \((5,4)\) as circle \(M\).
Using the standard form \((x - h)^2+(y - k)^2=r^2\), with \(h = 5\), \(k = 4\), \(r^2=36\), the equation is \((x - 5)^2+(y - 4)^2=36\).

Answer:

C. \((x - 5)^2+(y - 4)^2 = 36\)