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Vertices of Ellipse

A variable st...

Question

A variable straight line passes through the point $P (α, β)$ and cust the axes of coordinates in points $A$ and $B$ respectively. If the parallelogram $O A B C$ is completed then prove that the locus of vertex $C$ is $\frac{α}{x} + \frac{β}{y} = 1$

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Solution

$\frac{x}{a} + \frac{y}{b} = 1$ passes through $(α, β)$
$\frac{α}{a} + \frac{β}{b} = 1..... (1)$
Also $C$ is $(a, b)$ whose locus is $\frac{α}{x} + \frac{β}{y} = 1$ . Here $O A C B$ will be a rectangle and you may call it a parallelogram.

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