If the chord `xcosalpha+ysinalpha=p` of the hyperbola `(x^2)/(16)-(y^2)/(18)=1` subtends a right angle at the center, and the diameter of the circle, concentric with the hyperbola, to which the given chord is a tangent is `d ,` then the value of `d/4` is__________

Answer

24

Explanation

The equation of hyperbola is
`(x^(2))/(16)-(y^(2))/(18)=1`
`"or "9x^(2)-8y^(2)-144=0`
Homogenizing this equation using
`(x cosalpha+y sin alpha)/(p)=1`
We have `9x^(2)-8y^(2)-144((x cos alpha+y sin alpha)/(p))^(2)=0`
Since these lines are perpendicular to each other, we have
`9p^(2)-8p^(2)-144(cos^(2)alpha+sin^(2)alpha)=0`
`p^(2)=144or p = pm12`
`therefore" Radius of circle"=12`
`therefore" Diameter of circle"=24`