2014 AMC 12B Problem 17

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 \lt m \lt s.

What is

r + s?

1

26

40

52

80

The line is

y = m(x-20) + 14;

set it equal to

x^2

No intersection means the discriminant

m^2 - 80m + 56 \lt 0;

use the sum of the roots

r, s

The line through

Q

is

y = m(x-20) + 14.

Substituting into

y = x^2

gives

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

There is no intersection exactly when this has no real root, i.e. when the discriminant

m^2 - 4(20m-14)

= m^2 - 80m + 56

is negative. That happens between the two roots

r

and

s

of

m^2 - 80m + 56 = 0.

By Vieta’s formulas,

r + s = 80.

Thus, the correct answer is E.

Problem 17 in other years