# Thread: Need Help With a Tangent Line Problem

1. ## Need Help With a Tangent Line Problem

1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)

2) Determine where this line intersects the y-axis

2. Originally Posted by erimat89

1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)

2) Determine where this line intersects the y-axis
First find $y'$:

$y'=2Ax$.

Thus, the slope of the line at $x=c$ is $2Ac$.

Thus, the tangent has the equation $y-Ac^2=2Ac(x-c)\implies y=2Acx-Ac^2$

Now find where it crosses the x axis:

$0=2Acx-Ac^2\implies x=\dots$.

It crosses the y axis when x=0.

So $y=\dots$

Can you take it from here?

--Chris

Maybe I'm just confused but the question wasn't asking where it intersected the x-axis the x-axis intersection is given at the point (c/2, 0) it was only asking to determine where it crossed the y-axis in part 2 of the question. Why is this point (c/2, 0) given?

4. Originally Posted by erimat89

1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)

2) Determine where this line intersects the y-axis
Originally Posted by erimat89
Maybe I'm just confused but the question wasn't asking where it intersected the x-axis the x-axis intersection is given at the point (c/2, 0) it was only asking to determine where it crossed the y-axis in part 2 of the question. Why is this point (c/2, 0) given?
It asks you to show that the line intersects the x-axis at the point (c/2,0).

Since I got the equation of the line to be $y=2Acx-Ac^2$, at y=0, the line crosses the x-axis.

Thus $0=2Acx-Ac^2\implies 2Acx=Ac^2\implies x=\frac{Ac^2}{2Ac}\implies x=\frac{c}{2}$. So we see that it crosses the x-axis at $\left(\tfrac{1}{2}c,0\right)$

$\mathbb{Q.E.D.}$

Now, it intersects the y axis when x=0. Thus, we see that $y=2Ac(0)-Ac^2\implies y=-Ac^2$.

Thus, the y intercept is $\left(0,-Ac^2\right)$.

Does this make sense?

--Chris

5. ## Thanks

Makes alot of sense you clarified things for me. I've never been good with any sort of math problem that involves words for some reason I get thrown off.