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A point P moves such that the chord of c...

A point `P` moves such that the chord of contact of the pair of tangents from `P` on the parabola `y^2=4a x` touches the rectangular hyperbola `x^2-y^2=c^2dot` Show that the locus of `P` is the ellipse `(x^2)/(c^2)+(y^2)/((2a)^2)=1.`

Text Solution

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The equation of the chord of contact of parabola w.r.t. `P(x_(1),y_(1))` is given by
`yy_(1)=2a(x+x_(1))" (1)"`
Equation (1) touches the curve
`x^(2)-y^(2)=c^(2)" (2)"`
Using the condition of tangency on
`y=(2ax)/(y_(1))+(2ax_(1))/(y_(1))`
we get
`(4a^(2)x_(1)^(2))/(y_(1))=c^(2)(4a^(2))/(y_(1)^(2))-c^(2)" "[because"(1) is tangent to"(x^(2))/(c^(2))-(y^(2))/(c^(2))=1]`
`"or "4a^(2)x_(1)^(2)=4a^(2)c^(2)-c^(2)y_(1)^(2)`
`"or "4a^(2)x_(1)^(2)+c^(2)y_(1)^(2)=4a^(2)c^(2)`
Hence, the locus is
`(x^(2))/(c^(2))+(y^(2))/((2a)^(2))=1`
which is an ellipse.
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