Provide an appropriate response.
-Find the normal to the curve $x ^ { 2 } + y ^ { 2 } = 2 x + 2 y$ that is parallel to the line $y + x = 0$.
A) $y = - x + 2$
B) $y = - x - 2$
C) $y = x - 2$
D) $y = x + 2$

Question

Quizplus · Accepted Answer

The given curve is a circle, and the line $y + x = 0$ has a slope of -1. The normal to the curve will have the same slope as this line. Rearranging the equation of the circle to standard form, we get $x^2 - 2x + y^2 - 2y = 0$, which simplifies to $(x-1)^2 + (y-1)^2 = 2$, indicating a circle centered at (1,1) with radius $\sqrt{2}$. The normal line parallel to $y + x = 0$ and passing through the center (1,1) of the circle is $y = -x + 2$, which matches option A.

Provide an Appropriate Response x2+y2=2x+2yx ^ { 2 } + y ^ { 2 } = 2 x + 2 yx2+y2=2x+2y

Provide an Appropriate Response $x ^ { 2 } + y ^ { 2 } = 2 x + 2 y$