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Prove that the locus of the point of int...

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola `x^2-y^2=a^2` is `a^2(y^2-x^2)=4x^2y^2dot`

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Normal at a point `(a sec theta, a tan theta)` is
`x cos theta+y cot theta=2a`
If `P(x_(1),y_(1))` is the point of intersection of the tangents at the ends of normal chord (1), then (1) must be the chord of contact of P(h, k) whose equation is given by
`hx-ky=a^(2)" (2)"`
Comparing (1) and (2) and eliminating `theta`, we get
`(a^(2))/(4h^(2))-(a^(2))/(4k^(2))=1`
Hence, the locus is
`(1)/(x^(2))-(1)/(y^(2))=(4)/(a^(2))`
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