How to solve this ellipse question?

Question

"Show that S(sqrt(3),0) is a focus of the ellipse with equation 3x^2 + 4y^2 = 36."
I can do this, but then it goes on:
"The origin is O and P is any point on the ellipse. A line is drawn from O perpendicular to the tangent to the ellipse at P and this line meets the line SP, produced if necessary, in the point Q. Show that the locus of Q is a circle."
What I've tried:
~ Find the gradient of the tanget in terms of theta.
~ Use this to calculate the gradient of OQ (since this is perpendicular), and hence the equation of OQ in y, x, and theta (it goes through (0,0)).
~ Find the equation of PQ by finding the gradient and knowing it goes through point P, which has coordinates (2sqrt(3)cos(theta),3sin(theta)).
~ With both OQ and PQ as "y=..." they are equated to find the x coordinate of Q in terms of theta.
This is where I gave up because the result looked pretty horrible and I hadn't even worked out the y coordinate in terms of theta, never mind proved it was a circle. Any help?

Answer 1

Why don't you try using x, y cordinates instead?

Let P = (X,Y) we know the relation between X and Y
Find the gradient of the elipse at P and hence, the equation of the perpendicular passing through the origin.

Then find the equation for SP in terms of X and Y.

Solve simultaneously to find the cordinates of Q (x,y) in terms of X and Y.

Show that x^2 + y^2 (cords of Q) = constant independent of X and Y.

Answer 2

1) x^2/12 +y^2/9=1
so a^2=12 and b^2=9
c^2=a^2-b^2 =3 so c=+-sqrt3
If (Xo,Yo) is a point P The tangent at this point can be written as

(xXo)/12+yYo/9= 1 (a)
The perpendicular through O is
y=4/3(Yo/Xo)x

3yXo-4xYo=0 (b)
From (a) and b) you can calculate Xo and Yo depending onx and y and then put these values in the equation of the ellipse.
This should work

Answer 3

I think the BEST way to do this is to download the program called geometers sketchpad, I believe it has a free trial, and if you put the ellipse into it and draw some lines, it can calculate the angles and stuff, maybe that might help you if you can visuallise the prob

If you can't be bothered to learn how to use GSpad (which i think is very cool) then maybe download graphcalc which is FREE that is good too but is much less interactive.... hope that helps #