Two values for m for which a line is tangent to a parabola

1. Jul 20, 2014

rakeru

1. The problem statement, all variables and given/known data

Hi! I am doing some problems to practice for a math competition, and I'm wondering if I did this correctly. I don't really have an answer sheet, so I have no way of knowing whether I'm right. If you would please review it, that would be cool!

There are exactly two values of m for which the line y=mx-12 is tangent to the parabola y=x^2-3x+5. Determine those two values.

2. Relevant equations

y=mx-12
y=x^2-3x+5

3. The attempt at a solution

Okay so first I found the derivative of y=x^2-3x+5.

y'=2x-3

This is the slope, so I put it into the equation of the tangent line:

y=mx-12=(2x-3)x-12=2x^2-3x-12 <-- the tangent line

Now I made the the two equations equal to each other because they intersect at one point, right?

so:

x^2-3x+5=2x^2-3x-12

and when I solve for x it gives me +/- √17

I put this into the equation for the slope and I get:

m= 2√17 -3 or m=-2√17 -3

Is this correct? Thank you in advance.

2. Jul 20, 2014

Nathanael

Looks correct to me

3. Jul 21, 2014

HallsofIvy

Staff Emeritus
Another way to do it (used by Fermat previous to the development of calculus):
To find where the line y= mx- 12 crosses the parabola $y= x^2- 2x+ 5$, solve the equation $mx- 12= x^2- 2x+ 5$ or $x^2- (m+ 3)x+ 17= 0$

The "discriminant" of that equation is $(m+3)^2- 4(1)(17)= m^2+ 6m+ 9- 68= m^2+ 6x- 59$. The line will not touch the parabola at all if the equation has no real roots- if the discriminant is negative. It will cross through the parabola if the equation has two real roots- if the discriminant is positive. It will touch the parabola once (be tangent to the parabola) if the equation has a single root- if the discriminant is zero.

Thus, the line y= mx- 12 will be tangent to the parabola $y= x^2- 2x+ 5$ if and only if $m^2+ 6m- 59= 0$. That is the same as the equation you got and, yes, it has roots $-3\pm 2\sqrt{17}$.

4. Jul 21, 2014

rakeru

Thanks!

5. Jul 21, 2014

rakeru

Ouuuu interesting. Thank you :)