What is the center of the circle, given that its equation is x² + y² - 6x + 4y = 36?

A. (-3, 2)

The center of the circle given by the equation x 2 + y 2 − 6 x + 4 y = 36 is (3, -2), which is obtained by rewriting the equation in standard form. The provided answer option A (-3, 2) is incorrect. Thus, the correct center is (3, -2).
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Answered by Anonymous | 2025-07-04

To find the center of the circle given the equation x 2 + y 2 − 6 x + 4 y = 36 , we need to rewrite this equation in 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 of the circle and r is the radius.

Complete the Square :
We will complete the square for the x and y terms in the given equation.

Start with x 2 − 6 x :
To complete the square, take half of the coefficient of x , which is − 6 , divide by 2 to get − 3 , and then square it to get 9.
So, x 2 − 6 x can be rewritten as ( x − 3 ) 2 − 9 .

Now, for y 2 + 4 y :
Take half of the coefficient of y , which is 4, divide by 2 to get 2, and then square it to get 4.
So, y 2 + 4 y can be rewritten as ( y + 2 ) 2 − 4 .



Rewrite the Equation :
Substitute the completed squares back into the equation:
( x − 3 ) 2 − 9 + ( y + 2 ) 2 − 4 = 36 .
Simplify by combining constants:
( x − 3 ) 2 + ( y + 2 ) 2 = 36 + 9 + 4 .
( x − 3 ) 2 + ( y + 2 ) 2 = 49 .

Identify the Center :
The equation is now in the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , where h = 3 and k = − 2 .
Therefore, the center of the circle is ( 3 , − 2 ) .


Given the choices, option A. ( − 3 , 2 ) is incorrect. The correct answer should be ( 3 , − 2 ) .

Answered by EmmaGraceJohnson | 2025-07-06