How can I solve this? And what does it mean?

Question

Given the circle (x-6)^2 + (y+7)^2=225. Determine the center, length of the radius, and exact Circumference.

Answer 1

Question: Given the circle (x - 6)² + (y + 7)² = 225. Determine the center, length of the radius, and exact Circumference.

Answer: And Round 1 begins. Ha, round...

So here is the standard form for the equation of a circle:

(x - h)² + (y - k)² = r², where (h,k) represents the center of the circle, and r is the radius of the circle.

Center: Look at the standard form again. There are no plus (+) signs inside the parentheses, only minus (-) signs. So you need to be careful with how you read off the center from your equation. Rewrite the parenthetical terms in terms of subtraction:

(x - 6)² + (y - (-7))² = 225. Only one change needed. Now you can read off the center directly. h=6 and k=-7, so the center of the circle is (6, -7).

Radius: Look at the standard form again, and see how the right-side says r². Don't read the right-hand side of 225 immediately as the radius; you need to take the square root of 225 to get the radius).

Re-write the right-hand side in terms of a number being squared:

(x - 6)² + (y - (-7))² = (15)². Now you can read off the radius directly. r=15, so the length of the radius of the circle is 15.

Exact circumference: You have the radius now. Use the formula for circumference when radius is known:

C = 2πr --> C = 2π(15) --> C = 30π units.

Answer 2

Equation of circle is of form:

(x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.

Thus for the equation of the circle you are given:

(x-6)^2 + (y+7)^2=225, which is same as:

(x-6)^2 + (y - (-7))^2 = (15)^2

So per the definition of the equation of a circle:

(h,k) = (6,-7), which is the center of the circle

r = 15

The circumference of a circle is 2 * PI * radius= 94.2