The equation of the straight lines which are both tangent and normal to the curve `27x^(2)=4y^(3)` are

Options

A. `x= pm sqrt2(y-2)`
B. `x=pm sqrt3(y-2)`
C. `x=pmsqrt2(y-3)`
D. `x=pmsqrt3(y-3)`

Answer

A

Explanation

`x=2t^(3), y=3t^(2)`
`therefore" tangent at t is "x=yt=-t^(3)`
and normal at `t_(1)` is, `xt_(1)+y=2t_(1)^(4)+3t_(1)^(2)`
Since both equation represent same straight line,
`(1)/(t_(1))=-t=(-t^(3))/(2t_(1)^(4)+3t_(1)^(2))`
`rArr" "t^(6)-3t^(2)-2=0 rArr t^(2)=2 rArr t=pmsqrt2`
`therefore" lines are "x= pm sqrt2(y-2)`