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A tangent to the hyperbola x^(2)-2y^(2)=...

A tangent to the hyperbola `x^(2)-2y^(2)=4` meets x-axis at P and y-aixs at Q. Lines PR and QR are drawn such that OPRQ is a rectangle (where O is origin).Find the locus of R.

Text Solution

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Equation of tangent at point `(a sec theta, b tan theta) ` is
`(x)/(a) sec theta-(y)/(b) tan theta=1`
It meets axis at `P(a cos theta, 0 ) and Q(0, -b cot theta).`
Now, rectangle OPRQ is completed.
Let the coordinates of point R be (h, k).
`therefore" "h=a cos theta and k=-bcot theta`
`therefore" "sectheta=(a)/(h) and tan theta=(-b)/(k)`
Squaring and subtracting, we get `(a^(2))/(h^(2))-(b^(2))/(k^(2))=1.`
So, required locus is `(4)/(x^(2))-(2)/(y^(2))=1`.
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