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Let P be a point on the hyperbola x^2-y^...

Let `P` be a point on the hyperbola `x^2-y^2=a^2,` where `a` is a parameter, such that `P` is nearest to the line `y=2xdot` Find the locus of `Pdot`

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Consider any point P`(a sec theta, a tan theta)" on "x^(2)-y^(2)=a^(2)` This point will be nearest to y = 2x if the tangent at this point is parallel to y = 2x.
Differentiating `x^(2)-y^(2)=a^(2)` w.r.t. x, we get
`(dy)/(dx)=(x)/(y)`
`therefore" "[(dy)/(dx)](a sec theta, a tan theta)="cosec" theta`
The slope of y = 2x is 2. Therefore,
"cosec" `theta=2 or theta=(pi)/(6)`
Thus, `P-=(a sec.(pi)/(6),a tan.(pi)/(6))`
`-=((2a)/(sqrt3),(a)/(sqrt3))-=(h,k)`
`"i.e., "h=(2a)/(sqrt3)and k=(a)/(sqrt3)or k=(h)/(2)`
So, the required locus is `2y-x=0`.
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