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

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

First we need the derivative of the parabola, which is 2x + 1. Let (a, b) be a point on the parabola with tangent line passing through (2,-3). We have:

(b + 3)/(a - 2) = 2a + 1 (Using slope of a line and derivative of parabola)

b = a² + a (From parabola equation)

Solving for b in the first equation, we have:

b = (2a + 1)(a - 2) - 3

Substituting into the second equation, we have:

(2a + 1)(a - 2) - 3 = a² + a

2a² - 3a - 5 = a² + a

a² - 4a - 5 = 0

(a - 5)(a + 1) = 0

a = -1 , 5

Since b = a² + a, our two tangent points are (-1,0) and (5, 30). The slope at each respective point is:

2(-1) + 1 = -1

2(5) + 1 = 11

Using the point-slope form, our tangent lines are therefore:

y = -x - 1

y = 11(x - 2) - 3 = 11x - 25