If `x = 3 tan t` and `y = 3 sec t` then the value of `(d^2y)/(dx^2)` at `t = pi/4` , is `1/(ksqrt2)` . The value of k is _________.

Answer

6

Explanation

`x=3tant" "therefore" "y=3sect`
`y^(2)-x^(2)=9`
`2yy'-2x=0`
`y'=(x)/(y)`
`y'=(y-xy')/((y)^(2))`
`y'|_(t=(pi)/(4))=(3tan((pi)/(4)))/(3 sec((pi)/(4)))=(1)/(sqrt2),x=3, y=3sqrt2`
`y''=(3sqrt2-3sqrt2)/(18)=(3)/(18sqrt2)=(1)/(6sqrt2)`