The points on the curve 9y^{2} = x^{3} where | vedclass

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

Question

(C) The equation of the given curve is $9y^{2} = x^{3}$.Differentiating with respect to $x$,we have:$9(2y) \frac{dy}{dx} = 3x^{2} \Rightarrow \frac{dy

Vedclass · Accepted Answer

$\left( 4, \pm \frac{8}{3} \right)$

Vedclass · Answer

(C) The equation of the given curve is $9y^{2} = x^{3}$.Differentiating with respect to $x$,we have:$9(2y) \frac{dy}{dx} = 3x^{2} \Rightarrow \frac{dy}{dx} = \frac{3x^{2}}{18y} = \frac{x^{2}}{6y}$.The slope of the normal to the curve at point $(x_{1}, y_{1})$ is $-\frac{1}{\left(\frac{dy}{dx}ight)_{(x_{1}, y_{1})}} = -\frac{6y_{1}}{x_{1}^{2}}$.The equation of the normal at $(x_{1}, y_{1})$ is $y - y_{1} = -\frac{6y_{1}}{x_{1}^{2}}(x - x_{1})$.Rearranging into intercept form $\frac{x}{a} + \frac{y}{b} = 1$:$x_{1}^{2}(y - y_{1}) = -6y_{1}(x - x_{1})$$6y_{1}x + x_{1}^{2}y = 6x_{1}y_{1} + x_{1}^{2}y_{1} = y_{1}(6x_{1} + x_{1}^{2})$.Dividing by $y_{1}(6x_{1} + x_{1}^{2})$:$\frac{x}{\frac{y_{1}(6x_{1} + x_{1}^{2})}{6y_{1}}} + \frac{y}{\frac{y_{1}(6x_{1} + x_{1}^{2})}{x_{1}^{2}}} = 1$.Since the normal makes equal intercepts,the denominators must be equal:$\frac{6x_{1} + x_{1}^{2}}{6} = \frac{y_{1}(6x_{1} + x_{1}^{2})}{x_{1}^{2}}$.Assuming $6 + x_{1} 
eq 0$,we get $\frac{x_{1}}{6} = \frac{y_{1}}{x_{1}} \Rightarrow x_{1}^{2} = 6y_{1}$.Substituting $y_{1} = \frac{x_{1}^{2}}{6}$ into the curve equation $9y_{1}^{2} = x_{1}^{3}$:$9 \left(\frac{x_{1}^{2}}{6}ight)^{2} = x_{1}^{3} \Rightarrow 9 \cdot \frac{x_{1}^{4}}{36} = x_{1}^{3} \Rightarrow \frac{x_{1}^{4}}{4} = x_{1}^{3}$.Since $x_{1} 
eq 0$,$x_{1} = 4$.Then $y_{1}^{2} = \frac{4^{3}}{9} = \frac{64}{9} \Rightarrow y_{1} = \pm \frac{8}{3}$.Thus,the points are $\left( 4, \pm \frac{8}{3} ight)$.

The points on the curve $9y^{2} = x^{3}$ where the normal to the curve makes equal intercepts with the axes are

Similar Questions

Which microorganism is used for the commercial production of citric acid?

Which of the following is a micronutrient for plants?

Which of the following metal carbonates decomposes upon heating?

Two capacitors of capacities $2C$ and $C$ are joined in parallel and charged up to potential $V$. The battery is removed,and the capacitor of capacity $C$ is filled completely with a medium of dielectric constant $K$. The potential difference across the capacitors will now be

Glucose and mannose are

Vedclass Test Series

Exam Paper Generator

Online Exam Module