The numerical values of the volume and the lateral surface area of a right circular cone are equal. If the height and the radius of the cone are h unit and r unit respectively , then find the value of `(1)/(h^2)+(1)/(r^2)`.

Explanation

Let the slant height of the right circular cone is l unit . Also the height and radius of the cone are h and r unit respectively.
As per question , `(1)/(3)pi r^2h=pi rl`
or `rh=3lrArr r^2h^2=9 l ^2rArr r^2h^2=9 (h^2+r^2)`
`rArr (h^2+r^2)/(r^2h^2)=(1)/(9)rArr (h^2)/(r^2h^2)+(r^2)/(r^2h^2)=(1)/(9)rArr (1)/(r^2)+(1)/(h^2)=(1)/(9)rArr (1)/(h^2)+(1)/(r^2)=(1)/(9)`
Hence the value of `(1)/(h^2)+(1)/(r^2)=(1)/(9)`.