# Tangent lines to an ellipse from an external point

• Sep 4th 2012, 03:45 AM
Kaloda
Tangent lines to an ellipse from an external point
Given an the equation of an ellipse, say $\displaystyle \frac{x^2}{9}+\frac{y^2}{16}=1$,
find the equations of the two lines tangent to this ellipse passing through the external point
• Sep 4th 2012, 05:54 AM
topsquark
Re: Tangent lines to an ellipse from an external point
Quote:

Originally Posted by Kaloda
Given an the equation of an ellipse, say $\displaystyle \frac{x^2}{9}+\frac{y^2}{16}=1$,
find the equations of the two lines tangent to this ellipse passing through the external point

1. Solve the equation for y^2
2. Implicit differentiation

That gives the equation for the tangent lines.
3. Plug in point P.
4. What can you come up with?

-Dan
• Sep 4th 2012, 05:58 AM
kalyanram
Re: Tangent lines to an ellipse from an external point
Compute the slope of the tangent at the point of contact say $\displaystyle (\alpha,\beta)$which is $\displaystyle \frac{dy}{dx} = \frac{-16\alpha}{9\beta}$. Now we have the slope of the line also as $\displaystyle \frac{6-\beta}{5-\alpha} = \frac{-16\alpha}{9\beta} \implies 16\alpha^2 + 9 \beta^2 = 9*6\beta+9*10\alpha$ from the equation of ellipse we have $\displaystyle 16\alpha^2 + 9 \beta^2 = 9*16=9*6\beta+9*10\alpha \implies 5\alpha+3\beta=8$ substitute in $\displaystyle \frac{\alpha^2}{9} + \frac{\beta^2}{16} = 1$ and solve for $\displaystyle \beta$ you will get two points on the ellipse, and hence you can get two lines.

~Kalyan.
• Sep 8th 2012, 08:36 AM
Kaloda
Re: Tangent lines to an ellipse from an external point