Two tangents to the curve f (x) = x² - 8x + 4 intersect at the point (5, -20). A straight line runs through the two tangent points.

Question

Determine the coordinates of the intersection of this line with the x-axis.

Please help me walk through this step by step.

Answer 1

All tangents to f(x) have a slope given by the first derivative: f^'(x) = 2x - 8 = m_tangent

Any point on f has coordinates (x , x² - 8x + 4). So, a tangent line containing (5 , -20) will have a slope given by

[x² - 8x + 4 + 20] / (x - 5) = 2x - 8 Cross multiply:

x² - 8x + 4 + 20 = 2x² - 18x + 40 Gather like terms and solve the quadratic:

x² - 10x + 16 = 0

(x - 8)(x - 2) = 0 ; x = 2 or 8

The tangent pts. on f(x) are therefore (2 , - 8) and (8 , 4). The line containing these 2 pts has slope = 2 and eqn y - 4 = 2(x - 8). Finally, its x-intercept occurs when y = 0: - 4 = 2(x - 8) and x = 6.