Hi,

You can probably tell by now that me and conic section/locus questions don't get on.

This time we are considering an ellipse with equation

$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and focii at S and S'.

Consider an arbitrary point P on the ellipse, I need to show that the line SP and the perpendicular from the center O to the tangent at P meet at G, and that the locus of G is a circle centre S with radius a.

Well the gradient of the tangent at a point P(a cos(t), b sin(t)) is $\displaystyle \frac{-b \cos t}{a\sin t}$ so the gradient of the perpendicular to the tangent is $\displaystyle \frac{a \sin t}{b \cos t}$, this passes through the origin and so the equation for this line is $\displaystyle y= \frac{a \sin t}{b \cos t} x$

Now the gradient of the line SP is $\displaystyle \frac{\Delta y}{\Delta x} = \frac{b \sin t}{a \cos t + ae}$. This passes through the point (-ae, 0) and so the equation of the line SP is

$\displaystyle y=\frac{b \sin t}{a \cos t + ae}(x+ae)$

Presumably I must solve these equations simultaneously and then eliminate t to find the required equation of the circle. Since there is a lone y on each equation I tried setting them equal but got a horrific mess with the algebra so I think there's a trick I might be missing

Any help on how to solve these equations simultaneously (or even if my method is correct!) would be very much appreciated thank you.