"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?
2007-04-22
00:19:07
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3 answers
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asked by
Anonymous
in
Mathematics