jmc

Question

Let P be the parabola with equation y=x^2 and let Q = (20, 14). There are real numbers r and s such that the line through Q with slope m does not intersect P if and only if r < m < s. What is r + s?

Accepted Answer

The equation of the line passing through

Q = (20, 14)

with slope

m

is

y - 14 = m (x - 20) .

Thus, we seek the values of

m

for which the system

y - 14 y = m (x - 20), = x^{2}

has no real solutions.

Substituting

y = x^{2}

into the first equation, we get

x^{2} - 14 = m (x - 20) .

Then

x^{2} - m x + (20 m - 14) = 0.

This equation has no real solutions when the discriminant is negative:

m^{2} - 4 (20 m - 14) < 0.

Then

m^{2} - 80 m + 56 < 0.

Thus,

r

and

s

are the roots of

m^{2} - 80 m + 56 = 0.

By Vieta's formulas,

r + s = 80 .