Use implicit differentiation to find the points of tangency

[ziddy83]

Please Help!

Ok, I am having problem with an Implicit differentiation problem...
Two tangent lines to the hyperbola $$9x^2 - y^2 =36$$ intersect at the y-axis.

Use implicit differentiation to find the points of tangency. Ok so i implicitly differentiated this function and i came up with y'= 9x/y...now...Im not sure on how to find the points of tangency. Do i have to set the two equations equal to each other and solve for y and x seperately? Please help, and help will be appreciated! thanks

 

[HallsofIvy]

You really don't need to worry about "two" tangent lines. Choose any point on the ellipse, find the tangent line and see where it intersects the y_axis.

Let (x₀, y₀) be the point of tangency. Then the slope of the tangent line is, as you found, 9x₀/y₀ and the equation of the tangent line is $y= \frac{9x_0}{y_0}(x- x_0)+ y_0$. At the y-axis, x= 0 so
$y= \frac{9x_0}{y_0}(-x_0)+ y_0= \frac{-9x_0^2+y_0^2}{y_0}$.

 

[ziddy83]

Thank you for your response! I was just confused on what to pick the points as. Thanks again