If angle subtended by any chord of a rectangular hyperbola at the centre is `alpha` and angle between the tangents at ends of chord is `beta`, then

Options

A. `alpha=2beta`
B. `beta=2alpha`
C. `alpha+beta+pi`
D. `alpha+beta=(pi)/(2)`

Explanation

Let `P(ct_(1),(c )/(t_(1)))` and `Q(ct_(2),(c )/(t_(2)))` be two points on the hyperbola `xy=c^(2)`. Then,
`tanalpha=((1)/(t_(1)^(2))-(1)/(t_(2)^(2)))/(1-(1)/(t_(1)^(2))xx(1)/(t_(2)^(2)))` [ `:'` Slopes of `OP` and `OQ` are `(1)/(t_(1)^(2))` and `(1)/(t_(2)^(2))` respectively]
`impliestanalpha=(t_(2)^(2)-t_(1)^(2))/(t_(1)^(2)t_(2)^(2)-1)`
Slopes of tangents at `P` and `Q` are `-(1)/(t_(1)^(2))` and `-(1)/(t_(2)^(2))` respectively.
`:.tanbeta=(-(1)/(t_(1)^(2))+(1)/(t_(2)^(2)))/(1-(1)/(t_(1)^(2))xx(1)/(t_(2)^(2)))impliestanbeta=(t_(1)^(2)-t_(2)^(2))/(t_(1)^(2)t_(2)^(2)-1) `
Clearly, we have
`tanalpha=-tanbeta`
`impliestanalpha=tan(pi-beta)impliesalpha=pi-betaimpliesalpha+beta=pi`