### Home > APCALC > Chapter 7 > Lesson 7.2.1 > Problem7-58

7-58.

What is the value of $a$ such that the parabola $y = ax^2 + 3$ is tangent to the line $y = 5 - x$? Also, state the point of tangency.

At the point of tangency, $ax^2 + 3 = 5 − x.$

Since the tangent line is $y = 5 − x$, then the slope of the tangent is $−1$ at the point of tangency.
Therefore $2ax = −1$.

Solve a system of equations.