The product of perpendicular distances from any point on the hyperbola `9x^(2)-16y^(2)=144` to its asymptotes, is

Options

A. `(a^(2)b^(2))/(a^(2)+b^(2))`
B. `(ab)/(a^(2)+b^(2))`
C. `(a^(2)b^(2))/(a+b)`
D. `(1)/(a^(2)+b^(2))`

Explanation

Let `(asec theta, b tan theta)` be any point on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`.
The equation of the asymptotes of the given hyperbola are
`(x)/(a)-(y)/(b)=0` and `(x)/(a)+(y)/(b)=0`
Now,
`p_(1)=` Length of the perpendicular from `(a sec theta, b tan theta)`
on `(x)/(a)-(y)/(b)=0`
`impliesp_(1)=|(sec theta-tantheta)/(sqrt((1)/(a^(2))+(1)/(b^(2))))|`
and,
`p_(2)=` Length of the perpendicular from `(a sec theta, b tan theta)`
on `(x)/(a)+(y)/(b)=0`
`impliesp_(2)=|(sec theta+tantheta)/(sqrt((1)/(a^(2))+(1)/(b^(2))))|`
`:.p_(1)p_(2)=|(sec^(2) theta-tan^(2)theta)/((1)/(a^(2))+(1)/(b^(2)))|=(a^(2)b^(2))/(a^(2)+b^(2))`