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Conics Question (1 Viewer)

kev-kun

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P is a point on the ellipse x^2/a^2 + y^2/b^2 =1 with centre O. line drawn from O, parallel to the tangent to the ellipse at P, meets the ellipse at Q. Prove that the area of the triangle OPQ is independent of the position P.

Found this question quite tough :/ Any help is appreciated ^^
 

Trebla

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P is a point on the ellipse x^2/a^2 + y^2/b^2 =1 with centre O. line drawn from O, parallel to the tangent to the ellipse at P, meets the ellipse at Q. Prove that the area of the triangle OPQ is independent of the position P.

Found this question quite tough :/ Any help is appreciated ^^
Let the points have the parametric coordinates and .

I will use the perpendicular distance approach. The length of OP is given by



The equation of OP is given by



Hence the perpendicular distance (denoted as d) between Q and OP is given by



The area of the triangle (denoted as A) is therefore given by



BUT we haven't yet used the fact that OQ is parallel to the tangent at P. This gives the condition that



Therefore



which is independent of the positions of P (and Q)
 
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kev-kun

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Ohh thanks for the help ^^ But just one question what did you mean in:
BUT we haven't yet used the fact that OQ is parallel to the tangent at P. This gives the condition that
Not sure what you did there.
 

Trebla

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Ohh thanks for the help ^^ But just one question what did you mean in:
BUT we haven't yet used the fact that OQ is parallel to the tangent at P. This gives the condition that
Not sure what you did there.
I just equated the gradients of the tangent and the line OQ
 

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