Question

Given the equation of a circle below, find the diameter of the circle. Use only the digits 0-9 to enter the diameter.

[ x^2 + y^2 + 10x - 46y = -385 ]

Answer 1

To find the diameter of the given circle, first rewrite the equation in the standard form, complete the square, and determine the radius. The radius is found to be 13, so the diameter of the circle is twice the radius, which equals 26.

Step-by-step explanation:

To find the diameter of the circle represented by the equation x^2 + y^2 + 10x - 46y = -385, we need to rewrite the equation in the standard form of a circle's equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.

First, let's complete the square for both x and y terms:

1. Group the x terms together and the y terms together: (x^2 + 10x) + (y^2 - 46y) = -385.
2. Add 25 (which is (10/2)^2) to complete the square for the x terms and add 529 (which is (46/2)^2) to complete the square for the y terms on both sides of the equation: (x^2 + 10x + 25) + (y^2 - 46y + 529) = -385 + 25 + 529.
3. The equation now becomes (x + 5)^2 + (y - 23)^2 = 169.

From this, we can see that the radius r is the square root of 169, which is 13. Therefore, the diameter D (which is 2r) of the circle is 26.