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May 5th, 2009, 03:01 AM   #1
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Tangent Problem

If the line px + qy = r is a tangent to the ellipse a^2.x^2 + b^2.y^2 = c^2, then the maximum possible value of |pq| is

(A) max{a, b}|r|/c
(B) a^2b^2r^2/2c^4
(C) abr^2/c^2
(D) abr^2/2c^2.


Ans:

lx + my + n = 0 is a tangent of the ellipse x^2/a^2 + y^2/b^2 = 1 if n^2 = a^2.l^2 + b^2.m^2 .....(I)

Here the line is px + qy = r.

Eqation of the ellipse a^2x^2 + b^2y^2 = c^2

or, x^2/(c^2/a^2) + y^2/(c^2/b^2) = 1

So from (I), r^2 = (c^2/a^2).p^2 + (c^2/b^2).q^2

How the maximum possible value of |pq|will be found from above equation?


Thanks in advance!!
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