Cartesian equation of tangent of ellipse

**ssadi** · February 27th 2009, 05:30 AM

Show that for all values of m, the straight lines with equations $y=mx\pm \sqrt(b^2+a^2m^2)$ are tangents to the ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Heelp

**red_dog** · February 27th 2009, 11:23 AM

The system formed by the equation of the line and the equation of the ellipse has a single solution, therefore the line intersects the ellipse in a single point.

**HallsofIvy** · February 27th 2009, 03:18 PM

You might find it simpler to multiply on both sides of the equation to get $b^2x^2+ a^2y^2= a^2b^2$ so you can use implicit differentiation: $2b^2x+ 2a^2yy'= 0$ and $y'= -\frac{b^2x}{a^2y}$ .

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