# Need help with tangent lines and derivatives

• Oct 20th 2011, 09:43 PM
newslang
Need help with tangent lines and derivatives
Hey,

I'm struggling with this one problem from my text book.
"Find the equations of the tangent lines to the parabola y=(x^2)+x that pass through the point (2, -3)."

I know how to find the derivative (2x+1) and then I can find one set of points (5,30) but I can't figure out how to find the other set of points.

Thanks for your help!
• Oct 20th 2011, 09:51 PM
TheChaz
Re: Need help with tangent lines and derivatives
f'(a) = 2a + 1

And the slope of the line between the point (a, a^2 + a) and (2, -3) is (a^2 + a + 3)/(2 - a) = 2a + 1
Can you solve that?
• Oct 20th 2011, 09:59 PM
newslang
Re: Need help with tangent lines and derivatives
Yes, I get the points (5, 30) and (-1,0) and from there I can find the equations of the lines. Thank you so much!
• Oct 20th 2011, 10:07 PM
SammyS
Re: Need help with tangent lines and derivatives
Quote:

Originally Posted by newslang
Hey,

I'm struggling with this one problem from my text book.
"Find the equations of the tangent lines to the parabola y=(x^2)+x that pass through the point (2, -3)."

I know how to find the derivative (2x+1) and then I can find one set of points (5,30) but I can't figure out how to find the other set of points.

Thanks for your help!

Let's say you have a point (p,r) on your parabola. (So, r is given by $r=p^2+p\,.$ )

The slope, m, of the line tangent to the parabola when x = p is $m=2p+1$. (this from your derivative)

The slope, m, of the line passing through points (p, r) and (2, -3) is given by $m=\frac{r-\,-3}{p-2}=\frac{r+3}{p-2}\,.$

So that $m(p-2)=r+3\,.$

Plug the in the earlier results for r & m into this. Solve for p. (There will likely be two answers.) You then have the x value(s) for the intersection of the tanget line, with the parabola.
• Oct 21st 2011, 09:35 AM
HallsofIvy
Re: Need help with tangent lines and derivatives
Quote:

Originally Posted by newslang
Hey,

I'm struggling with this one problem from my text book.
"Find the equations of the tangent lines to the parabola y=(x^2)+x that pass through the point (2, -3)."

I know how to find the derivative (2x+1) and then I can find one set of points (5,30) but I can't figure out how to find the other set of points.

Thanks for your help!

Another way to do a problem like this is to use Fermat's "method of ad-equations" which predates Calculus. Any line through (2, -3) can be written y= m(x-2)- 3= mx- 2m- 3. Of course, at a point where such a line crosses the graph of $y= x^2+ x$ the two functions must give the same y-value: $mx- 2m- 3= x^2+ x$ or $x^2+ (1- m)x+ 2m+ 3= 0$. That's a quadratic equation with solution given by the quadratic formula: $x= \frac{m-1\pm\sqrt{(1- m)^2- 4(2m+ 3)}}{2}$. But if it is tangent that x must be a double root- the only root of this equation. That means that the discriminant must be 0: $(1- m)^2- 4(2m+3)= 0$. Solve that equation for m and then $x= \frac{m-1}{2}$.

A quadratic equation may have, of course, two roots so there may be two tangent lines that pass through (2, -3).