Question

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

• A. $x=\pm \sqrt{2}(y-3)$
• B. $x=\pm \sqrt{3}(y+2)$
• C. $x=\pm \sqrt{2}(y+2)$
• D. $x=\pm \sqrt{2}(y-2)$

Answer 1

The correct option is D $x=\pm \sqrt{2}(y-2)$

$x=2t^3,y=3t^2$
Tangent at ${}^{\prime }{t}^{\prime }$ $\to x-ty=-t^3$
Normal at ${}^{\prime }{t}_1^{\prime },\to t_{1}x+y=3t_1^2+2t_1^4$
So $t_1=-t=\frac{-t^3}{3t_1^2+2t_1^4}$
$-t=\frac{-t^3}{\frac{3}{t^2}+\frac{2}{t^4}}\Rightarrow t\ne 0,\frac{3}{t^2}+\frac{2}{t^4}=t^2$
$\Rightarrow 3t^2+2=t^6$
$t=\pm \sqrt{2}$
Tangent $x=t(y-t^2)\Rightarrow x=\pm \sqrt{2}(y-2)$