# Tangent to the ellipse

• September 29th 2010, 04:19 AM
ecogreen
Tangent to the ellipse
I need help with this maths question...

Prove that if the line $lx +my + n = 0$ touches the ellipse $b^2 x^2 + a^2 y^2 = a^2 b^2$ then $a^2 l^2 + b^2 m^2 = n^2$

I tried substituting the first equation in terms of x or y, then substitute into the second equation but I ended up having too many unknowns.... any ideas how to solve this question...

Thank you..
• September 29th 2010, 06:55 AM
earboth
Quote:

Originally Posted by ecogreen
I need help with this maths question...

Prove that if the line $lx +my + n = 0$ touches the ellipse $b^2 x^2 + a^2 y^2 = a^2 b^2$ then $a^2 l^2 + b^2 m^2 = n^2$

I tried substituting the first equation in terms of x or y, then substitute into the second equation but I ended up having too many unknowns.... any ideas how to solve this question...

Thank you..

1. Calculate the coordinates of the points of intersection. If the line is actually a tangent to the ellipse you only get one point of intersection.

2. $lx+my+n=0~\implies~y=-\dfrac{lx+n}{m}$

Plug in the term for y into the 2nd equation and solve for x.

3. $b^2 x^2+a^2 \cdot \left(-\dfrac{lx+n}{m}\right)^2=a^2 b^2$

Expand the bracket and collect like terms:

$\dfrac{a^2 l^2 +b^2 m^2}{m^2} \cdot x^2+\dfrac{2 a^2 l n}{m^2} \cdot x + \dfrac{a^2 n^2}{m^2} = a^2 b^2$

This is a quadratic equation in x. Use the quadratic formula and solve for x:
$x = \dfrac{-a^2 l n \pm \sqrt{a^2 l^2 + b^2 m^2 - n^2} }{a^2 l^2 + b^2 m^2}$

4. You only get one common point of the line and the ellipse (= tangent point) if the radical equals zero:

$a^2 l^2 + b^2 m^2 - n^2=0~\implies~\boxed{a^2 l^2 + b^2 m^2 = n^2}$