# x-intercept of tangent to parabola

• Jan 8th 2006, 09:18 AM
frozenflames
x-intercept of tangent to parabola
Show that the tangent to the parabola Y=Ax^2 (for A does not equal 0) at the point where x = c will intersect the x axis at the point (c/2 , 0 . Where does it intersect the y axis.
• Jan 8th 2006, 10:07 AM
ThePerfectHacker
Quote:

Originally Posted by frozenflames
Show that the tangent to the parabola Y=Ax^2 (for A does not equal 0) at the point where x = c will intersect the x axis at the point (c/2 , 0 . Where does it intersect the y axis.

To find the tangent find the derivative of $y=ax^2$ at $x=c$ which is $\lim_{h\rightarrow 0} \frac{a(x+h)^2-ax^2}{h}=2ac$
Thus, the slope of the tangent line is $m=2ac$. Now use the formula for finding equations of line by knowing thier slopes and the point that they pass through. That formula is $y-y_0=m(x-x_0)$this tangent line has slope $m=2ac$ and passes through the point of tangentcy $x=c$ thus $y=ac^2$ Thus, the equation of the tangent line is $y-ac^2=2ac(x-c)$ thus, $y=2acx-2ac^2+ac^2=2acx-ac^2$ now set $y=0$ because this is the x-intercept thus $2acx-ac^2=0$ solve for $x$ thus $x=c/2$
Q.E.D.
• Jan 8th 2006, 10:16 AM
frozenflames
Where does it intersect the y Axis?
• Jan 8th 2006, 10:22 AM
ThePerfectHacker
Quote:

Originally Posted by frozenflames
Where does it intersect the y Axis?

Easy since the equation of the tangent line is $y=2acx-ac^2$ the y-intersept is when $x=0$ thus, $y=-ac^2$
Q.E.D.