Calculus III Problem

Question

At what angle does the circle x²+y²=2 intersect with the parabola y=x²?

Answer 1

x^2 + y^2 = 2

intersects y=x^2

when y^2 +y -2 = 0

(y+2)(y-1) = 0

y =1, -2

x = 1

they intersect at the point

(1,1)

slope of the curves = y'=2x = 2 for theparabola

slope of the circle: 2yy' = -2x

y' =-x/y = -1/1 = -1

a tangent line with slope 2 intersects with a tangent line of slope =-1 at an angle = 60+45 = 105 degrees

slope -1 means -45 degrees with the horizontal, slope 2 means +60 degrees with the horizontal

60- -45 = 105 degrees