Hey!

I didn't really know what to put as a title so I hope it describes the problem relatively well... I have been working on this problem for about 20 minutes and haven't come very far, so I was hoping for someones help in putting me in the right direction!

Question

For what values of a and b is the line $\displaystyle 2x + y = b $ tangent to the parabola $\displaystyle y = ax^2 $ when x = 2?

Solution

I feel I have all the pieces of the puzzle, but there is some trick to putting it together.

I know how to find the equation of a tangent line when given an equation and a point on that equation (just find the derivative and then use y = mx + b and sub in everything to find b). However in this question I don't even know where to start. I know that $\displaystyle y' = 2ax $.

If I sub 2 into either y' or y I get y = 4a(or y' = 4a). How do I use this information to solve for a and b?

This feels like an easy problem that I just can't wrap my head around!

Thanks for your help!