x^2 + y^2 - 20x - 60y - 1500 = 0

To solve the equation x 2 + y 2 − 20 x − 60 y − 1500 = 0 , we can rewrite it in the form of a circle by completing the square. A circle in the coordinate plane is generally represented by the equation ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius.
Let's work through the steps to find the center and radius of the circle represented by this equation:

Group and rearrange the terms: x 2 − 20 x + y 2 − 60 y = 1500

Complete the square for the x terms:



Take half of the coefficient of x , which is − 20 , divide by 2 to get − 10 , and then square it to get 100.

Add and subtract 100 inside the equation: ( x 2 − 20 x + 100 )



Complete the square for the y terms:


Take half of the coefficient of y , which is − 60 , divide by 2 to get − 30 , and then square it to get 900.

Add and subtract 900 inside the equation: ( y 2 − 60 y + 900 )



Reorganize and simplify the equation:


Now our equation looks like this: ( x − 10 ) 2 + ( y − 30 ) 2 = 1500 + 100 + 900 ( x − 10 ) 2 + ( y − 30 ) 2 = 2500


Identify the center and radius:


The equation ( x − 10 ) 2 + ( y − 30 ) 2 = 2500 is in the standard form of a circle.

The center of the circle is ( h , k ) = ( 10 , 30 ) .

The radius r is the square root of 2500, which is 50.


In summary, the equation describes a circle with a center at ( 10 , 30 ) and a radius of 50. Understanding the process of completing the square is crucial in transforming quadratic equations into a recognizable geometric form.

Answered by JessicaJessy | 2025-07-06

The equation x 2 + y 2 − 20 x − 60 y − 1500 = 0 can be rewritten as a circle's equation by completing the square. The center of the circle is at ( 10 , 30 ) and the radius is 50. This transformation is key for understanding geometric representations of equations.
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Answered by JessicaJessy | 2025-07-08