The equations of straight lines which are bot | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given curve: $27x^2 = 4y^3$.Differentiating with respect to $x$: $54x = 12y^2 \frac{dy}{dx} \Rightarrow \frac{dy}{dx} = \frac{9x}{2y^2}$.Let the p

Vedclass · Accepted Answer

$x = \pm \sqrt{2}(y - 2)$

Vedclass · Answer

(D) Given curve: $27x^2 = 4y^3$.Differentiating with respect to $x$: $54x = 12y^2 \frac{dy}{dx} \Rightarrow \frac{dy}{dx} = \frac{9x}{2y^2}$.Let the point on the curve be $(2t^3, 3t^2)$.Slope of the tangent at $t$: $m = \frac{9(2t^3)}{2(3t^2)^2} = \frac{18t^3}{18t^4} = \frac{1}{t}$.Equation of the tangent at $t$: $y - 3t^2 = \frac{1}{t}(x - 2t^3) \Rightarrow x - ty = -t^3$  .......$(i)$.Equation of the normal at $t_1$: $y - 3t_1^2 = -t_1(x - 2t_1^3) \Rightarrow t_1x + y = 3t_1^2 + 2t_1^4$  .......$(ii)$.Since the line is both tangent and normal,comparing $(i)$ and $(ii)$ gives $\frac{1}{t_1} = -t = \frac{-t^3}{3t_1^2 + 2t_1^4}$.From $\frac{1}{t_1} = -t$,we have $t_1 = -\frac{1}{t}$.Substituting into $-t = \frac{-t^3}{3t_1^2 + 2t_1^4}$: $t = \frac{t^3}{3(\frac{1}{t^2}) + 2(\frac{1}{t^4})} = \frac{t^7}{3t^2 + 2}$.$3t^2 + 2 = t^6 \Rightarrow t^6 - 3t^2 - 2 = 0$. Let $u = t^2$,then $u^3 - 3u - 2 = 0$.$(u - 2)(u + 1)^2 = 0$. Since $u = t^2 > 0$,we have $t^2 = 2$,so $t = \pm \sqrt{2}$.Substituting $t^2 = 2$ into the tangent equation $x = t(y - t^2)$: $x = \pm \sqrt{2}(y - 2)$.

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

Similar Questions

What is the length of the normal at any point on the curve $y = \frac{1}{2}a(e^{x/a} + e^{-x/a})$?

If $\theta$ is the acute angle between the curves $y^2 = x$ and $x^2 + y^2 = 2$, then $\tan \theta =$

The angle between the curves $y^2=2x$ and $x^2+y^2=8$ is

Find the point on the curve $y = \sqrt{4x - 3} - 1$ where the tangent has a slope of $\frac{2}{3}$.

The points on the curve $y^2 = x + \sin x$ at which the normal is parallel to the $Y$-axis lie on

Vedclass Test Series

Exam Paper Generator

Online Exam Module