Question

this circle is centered at the point (4, 5), and the length of its radius is 3. what is the equation of the circle?
a. $(x + 4)^{2}+(y + 5)^{2}=3$
b. $(x - 5)^{2}+(y - 4)^{2}=9$
c. $(x - 4)^{2}+(y - 5)^{2}=9$
d. $(x^{2}-4)+(y^{2}-5)=3^{2}$

Explanation:

Step1: Recall the standard form of a circle's 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: Identify the center and radius values

Given the center \((h,k)=(4,5)\) and radius \(r = 3\).

Step3: Substitute the values into the standard form

Substitute \(h = 4\), \(k = 5\), and \(r^2=3^2 = 9\) into \((x - h)^2+(y - k)^2=r^2\), we get \((x - 4)^2+(y - 5)^2=9\).

Answer:

C. \((x - 4)^2+(y - 5)^2 = 9\)