Practicing Success
The points on the curve $9y^2=x^3$ where normal to the curve makes equal intercepts with the axes is/are : |
$\left(4, \frac{8}{3}\right),\left(4, \frac{-8}{3}\right)$ $\left(0, \frac{8}{3}\right)$ $\left(0, \frac{-8}{3}\right)$ $(4, 4)$ |
$\left(4, \frac{8}{3}\right),\left(4, \frac{-8}{3}\right)$ |
The correct answer is Option (1) → $\left(4, \frac{8}{3}\right),\left(4, \frac{-8}{3}\right)$ $9y^2=x^3$ ...(1) differentiating wrt (x) $18y\frac{dy}{dx}=3x^2$ so $\frac{dy}{dx}=\frac{x^2}{6y}$ slope of normal → $-\frac{dy}{dx}=(-\frac{x^2}{6y})^{-1}$ as it makes equal intersepts $-\frac{x^2}{6y}=-1⇒6y=x^2$ or $y=\frac{x^2}{6}$ from (1) $9×\frac{x^4}{36}=x^3$ so $x=4$ from (1) $9y^2=4^3$ $y^2=\frac{64}{9}$ $y=±\frac{8}{3}$ Points $\left(4, ±\frac{8}{3}\right)$ |