Question

Find equations of both lines through the point (2, −3) that are tangent to the parabola y = x2 + x.

y = (smaller slope)

y = (larger slope)

Answer 1

y = -x - 1 ; y = 11x - 25

Explanation:

y = x² + x

y-(-3) = m(x-2)

y = x²+x

y + 3 = mx - 2m

y = x² + x

y = mx - 2m - 3

x² + x = mx - 2m - 3

x² + x - mx + 2m + 3 = 0

x² + (1-m)x + 2m + 3 = 0

discriminant = m² + 1 - 2m -4 (2m + 3)

m² + 1 - 2m - 8m - 12

m² - 10m - 11

the line is tangent so the discriminant must be equal to 0

m² - 10m - 11 = 0
(m-11)(m+1) = 0

m = 11

m = -1

y + 3 = -(x-2)

y+ 3 = -x + 2

y = -x + 2 - 3

y = -x - 1

y + 3 = 11(x-2)

y + 3 = 11x - 22

y = 11x -22 - 3

y = 11x - 25