If four points be taken on a rectangular hyperbola such that the chord joining any two is perpendicular to the chord joining the other two, and if `alpha, beta, gamma, delta` be the inclinations of either asymptote of the straight line joining these points to the centre, prove that `tanalphatanbetatangammatandelta=1`

Options

A. `1`
B. `2`
C. `-2`
D. `-1`

Explanation

Let `xy=c^(2)` be the rectangular hyperbola, and let `A(ct_(1),(c )/(t_(1)))`, `B(ct_(2),(c )/(t_(2)))`, `C(ct_(3),(c )/(t_(3)))` and `D(ct_(4),(c )/(t_(4)))` be the four points on it such that `AB` is perpendicular to `CD`.

`:.((c )/(t_(2))-(c )/(t_(1)))/(ct_(2)-ct_(1))xx((c )/(t_(4))-(c )/(t_(3)))/(ct_(4)-ct_(3))=-1impliest_(1)t_(2)t_(3)t_(4)=-1`........`(i)`
Clearly, `OX` and `OY` and are asymptotes of `xy=c^(2)`.
It is given that `OA`, `OB`, `OC` and `OD` make angles `alpha`, `beta`, `gamma`, and `delta` respectively with `OX`.
`:.tanalpha=(1)/(t_(1)^(2))`, `tanbeta=(1)/(t_(2)^(2))`, `tangamma=(1)/(t_(3)^(2))` and `tan delta=(1)/(t_(4)^(2))`
`impliestanalphatanbetatangammatandelta=(1)/(t_(1)^(2)t_(2)^(2)t_(3)^(2)t_(4)^(2))=1` [Using `(i)`]