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
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 so the gradient of the perpendicular to the tangent is , this passes through the origin and so the equation for this line is
Now the gradient of the line SP is . This passes through the point (-ae, 0) and so the equation of the line SP is
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.