Question

Find an equation for the tangent lines to the hyperbola x^2-y^2 = 16 that pass through (2,-2).

Answer 1

I got the first part down. I'm just not sure how to find the equation of the line that goes through (2,-2) and touches the hyperbola

\[x^{2}+y^{2} = 16\]

\[\frac{ d }{ dx }x^{2}+\frac{ d }{ dx }y^{2} = 0 \]

\[2x+2y~\frac{ dy }{ dx } = 0 \]

\[\frac{ dy }{ dx } = \frac{ -2x }{ -2y }\]

\[\frac{ dy }{ dx } = \frac{ x }{ y }\]

Answer 2

how did you get the extra negative factor ?

Answer 3

err... wait.. sorry it was x^{2}-y^{2} = 16

Answer 4

ok lol the other thing was a circle anyways :p
that makes more sense

Answer 5

yeah, Like i don't know what to do after this.. it's saying something like find an equation for the tangent line that also passes through the point (2,-2)

Answer 6

so say the line goes through (a,f(a)) and (2,-2)
where (a,f(a)) is the point of tangent in question

Answer 7

yep following

Answer 8

\[a^2-(f(a))^2=16 \\ -(f(a))^2=16-a^2 \\ (f(a))^2=a^2-16 \\ f(a)= \pm \sqrt{a^2-16} \]
so we have two assumptions to make about f(a)
\[\text{ assume } f(a) >0 \\ \frac{dy}{dx}|_{(x,y)=(a,\sqrt{a^2-16})} =\frac{a}{\sqrt{a^2-16}} \\ \text{ but we also can find the slope of the line algebraically } \\ \text{ that is using } \frac{\sqrt{a^2-16}-(-2)}{a-2}\]

Answer 9

so we have an equation to solve

Answer 10

we will go back later and make the assumption y is negative

Answer 11

i mean f(a)

Answer 12

really ridiculous question: on the second line what happened to the negative sign?

Other question is this:

okay so you're saying that two points a and f(a) are on the circle.

and that we insert those points into the equation for a circle and then isolate to say find
one of the points?

Answer 13

hyperbola :p
also we are trying to find the point of tangency in question
a point of tangency will happen on the relation/function
so yes it is on the hyperbola

we are using the calculus definition of slope and using the algebraic definition of slope to solve for a

Answer 14

and one sec about the negative sign question

Answer 15

are you talking about going from 2nd to 3rd
or from 1st to 2nd?

Answer 16

1st to 2nd I subtract something on both sides
2nd to 3rd I multiplied both sides by -1

Answer 17

-(f(a))^{2} = 16-a^{2}

Answer 18

I'm going to draw a graph of what we are doing
I only drew one possible tangent line that contained (2,-2)
there looks like there is 3 more