One can draw a diagram to show that there are two tangent lines to the parabola 𝑦=𝑥2+4 that pass through the point (0,0).

Question

One can draw a diagram to show that there are two tangent lines to the parabola 𝑦=x^2+4 that pass through the point (0,0). Find the x-coordinates of the points where these tangent lines intersect the parabola. 

Enter these x-coordinates in ascending order below.

Answer 1

A tangent line at x=a has equation y = f(a) + f'(a) (x-a)

y = x^2 + 4 so f(x) = x^2 + 4 and f'(x) = 2x

so consider tangent lines through (a, a^2+4) and (0,0)

Tangent line equation y = a^2+4 + 2a (x-a)

When y=0 x = 0, so 0 = a^2 + 4 + 2a (-a) = a^2 -2a^2 + 4 = 4 - a^2

So a = +2 or a = -2

So the two points are (-2, 8) and (2, 8) and the x coordinates are -2, 2