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

Question

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

rogue · Answer

$$y = x^2 - 3$$$$\frac {dy}{dx} = 2x$$$$y' (1) = 2$$$$Y _{tangent}^{} = 2(x-1)-2$$ 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.

anonymous · Answer

Negative reciprocal! Geesh. That helps! Thanks!

rogue · Answer

Hehe, well, now you know :)

anonymous · Answer

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 :/

rogue · Answer

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

rogue · Answer

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

rogue · Answer

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).

rogue · Answer

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

anonymous · Answer

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

anonymous · Answer

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