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Show that the locus represented by x=(1)...

Show that the locus represented by `x=(1)/(2)a(t+(1)/(t)),y=(1)/(2)a(t-(1)/(t))` is a rectangular hyperbola.

Text Solution

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Let point P be `(a sec theta, b tan theta)`.
Equation of tangent at point P is `(x)/(a)sec theta-(y)/(b)tan theta=1`
Equation of asymptotes are `y=pm(b)/(a)x.`
Solving asymptotes with tangent, we get
`Q-=((a)/(sectheta-tantheta),(b)/(sectheta-tantheta))`
`"and "R-=((a)/(sectheta+tantheta),(-b)/(sectheta+tantheta))`
`therefore" Area of triangle CQR"=(1)/(2)||(0,0),((a)/(sectheta-tantheta),(b)/(sectheta-tantheta)),((a)/(sectheta+tantheta),(-b)/(sectheta+tantheta)),(0,0)||`
`=(1)/(2)|-(a)/(sectheta-tantheta).(b)/(sectheta+tantheta)-(a)/(sectheta-tantheta).(b)/(sectheta+tantheta)|`
= ab, which is constant.
Also, midpoint of QR is
`(((1)/(sectheta-tantheta)+(1)/(sectheta+tantheta))/(2),((b)/(sectheta-tantheta)-(b)/(sectheta+tantheta))/(2))`
`-=(a sec theta, b tan theta)`, which is point P.
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