Question

Let $A,$ $B$ and $C$ be three distinct points on the graph of $y = x^2$ such that line $AB$ is parallel to the $x$ -axis and $\triangle ABC$ is a right triangle with area $2008.$ What is the sum of the digits of the $y$ -coordinate of $C?$

$16$

$17$

$18$

$19$

$20$

Answer: C

Concepts:parabola slope triangle area digits

Difficulty rating: 1800

Solution:

Since $AB$ is horizontal, take $A = (a, a^2)$ and $B = (-a, a^2),$ and let $C = (c, c^2).$ The right angle cannot be at $A$ or $B$ (that would need $c = \pm a$ ), so it is at $C.$

Put $d = a^2-c^2.$ Taking the dot product of $\overrightarrow{CA}$ and $\overrightarrow{CB}$ gives $d(d-1)=0.$ The points are distinct, so $d \ne 0;$ hence $a^2-c^2=1.$ This value is the height of the triangle above $\overline{AB}.$

The area is $\tfrac12 \cdot AB \cdot \text{height}$ $= \tfrac12 (2|a|)(1)$ $= |a| = 2008,$ so $a^2 = 2008^2 = 4{,}032{,}064$ and the $y$ -coordinate of $C$ is $c^2 = a^2 - 1 = 4{,}032{,}063.$

Its digit sum is $4 + 0 + 3 + 2 + 0 + 6 + 3 = 18.$

Thus, the correct answer is C.

2008 AMC 12B Problem 17

Answer: C

Difficulty rating: 1800

Solution:

Problem 17 in Other Years