Question

Find an equation of the line tangent to the curve 2y^2 - x^2 =1 at the point (7,-5)
this is my answer
http://i.imgur.com/lwzCXHJ.png
after checking many times, I still can't get the right answer which is
y= (-7/10)x-(17/2)
Can anyone help me check where my problem is?

Answer 1

Your error is immediately apparent. The point is (7,-5) anf when you solve for y, you managed to discard all negative values for y. Check it out. Substitute any valid value for x and see if you can get a negative value for y from \(y = \sqrt{\dfrac{1+x^{2}}{2}}\).

There are a couple of ways that might fix this.

1) Solve for a version of y that allows for negative values. \(y = -\sqrt{\dfrac{1+x^{2}}{2}}\), but I don't really recommend that.

2) Use implicit differentiation and get them ALL without concerning yourself with the sign of anything. I demonstrate:

\(2y^{2} - x^{2} =1\)

\(4y\cdot y' - 2x\;=\;0 \implies y'\;=\;\dfrac{2x}{4y}\;=\;\dfrac{x}{2y}\)

If you are on a section about Implicit Differentiation, you should take advantage of teh method and not get all excited about algebra. Remember your algebra, but don't overuse it.

Answer 2

very detailed, thank you for your help ^^

Answer 3

It was a subtle error. Anyone could have missed it. Truthfully, I am a little surprised that it just jumped out at me.

ALWAYS think about the Domain and the Range. Always. It probably seemed so pointless when we made you think about it back in algebra class.

This is one reason why we studied the difference between a Function and a Relation. Functions are a little more reliable. Hyperbolas are not functions and they take a little more care.

Just for the record, your algebra and derivative were pretty good, just a little misguided. :-)