Find the intersections

Question

Find the intersections of the line y=-x-1 and the circle x^2+y^2=41. Show your work.

I graphed and found that the answers are (-5,4) and (4,-5), but I’m not sure how to solve this without graphing.

Answer 1

x² + y² = 41

y² = 41 - x²

For the line:

y = -x - 1

y² = (-x - 1)² = x² + 2x + 1

Now set the two y² terms equal to one another. You can do this because where they intersect, they are equal:

41 - x² = x² + 2x + 1

0 = 2x² + 2x - 40

0 = x² + x - 20 [divided both sides by 2]

0 = (x+5)(x-4) [factored the quadratic]

x = -5 and 4 ---> these are the two x-values

To find the y-values, use y = -x - 1:

For x = -5

y = 5-1 = 4

So one point is (-5,4)

For x = 4

y = -4 - 1 = -5

The other point is (4,-5)