## anonymous 4 years ago At what point does the normal line to the curve y = x^2-3 at (1, -2) intersect the curve a second time? The derivative is 2x so the slope at (1, -2) is 2 thus the equation of the tangent line is y+2 = 2 ( x-1) which I can graph both of, and am still not seeing where it crosses twice: http://www.wolframalpha.com/input/?i=y%2B2+%3D+2+%28+x-1%29+and+y+%3D+x%5E2-3

1. Rogue

$y = x^2 - 3$$\frac {dy}{dx} = 2x$$y' (1) = 2$$Y _{tangent}^{} = 2(x-1)-2$ The normal line to the curve is perpendicular to the curve, so it has the negative reciprocal slope of the tangent.$Y _{normal}^{} = \frac {-1}{2}(x-1)-2$So now just set Ynorm equal to y = x^2 - 3 to find the other coordinate. My tangent and normal line equations are in the linearization form by the way, you can simplify them down if you want.

2. anonymous

Negative reciprocal! Geesh. That helps! Thanks!

3. Rogue

Hehe, well, now you know :)

4. anonymous

err ok so I set -1/2(x-1)-2 = x^2-3 and eventually got x^2+1/2x - 3/2 which, when I put in 1 for x, I get 0 out which is evidently not correct. I guess I am still confused after all :/

5. Rogue

$Y _{norm}^{} = -0.5x -1.5$$Y = x^2 - 3$$0 = x^2 - 3 + 0.5x + 1.5 = x^2 + 0.5x - 1.5$ Solving for that should give you the other solution, which is -1.5. I think you just made a computation error somewhere, no biggie.

6. Rogue

Oh, you put 1 into that... No...

7. Rogue

When you set Ynorm = Y, you are finding when the 2 graphs intersect. Its not the derivative or anything, so you don't plug in 1. You already know that they intersect at the point given to you, (1,-2). You just have to find the other point. Solving for that equation will give you x = 1, x = -1.5. So the other coordinate where the normal & the parabola intersects has a x-coordinate of -1.5 Plugging in -1.5 into either equations will give us the y coordinate, which is -0.75. So the other point where the normal intersects the curve is (-1.5, -0.75).

8. Rogue

Do you get it, or need a bit more explanation?

9. anonymous

I'm sorry, reviewing now, had a kid wake up with a bad dream -_-

10. anonymous

Thank you!!! I /do/ understand!! I have no idea how I would've gotten that without your help.