Question

A curve is given as $y=x^3+3x^2+7$. The rate of change of abscissa at a certain point is equal to $\frac{1}{9}$ of the rate of change of ordinate. Which of the following denotes that point , if it is known to lie in $1^{st}$ Quadrant.

• A. $(2,11)$
• B. $(1,11)$
• C. $(2,27)$
• D. $(1,13)$

Answer 1

The correct option is B $(1,11)$

$y=x^3+3x^2+7$
$\frac{dy}{dt}=(3x^2+6x)\frac{dx}{dt}$
$\{\because \frac{dx}{dt}=\frac{1}{9}\frac{dy}{dt}\}$
$\Rightarrow 9\frac{dx}{dt}=(3x^2+6x)\frac{dx}{dt}$
$\Rightarrow x^2+2x-3=0$
$\Rightarrow (x-1)(x+3)=0$
$\Rightarrow x=1$ $\{\because x\ne -3\}$ as point lies in $1^{st}$Quadrant
$\Rightarrow y=1+3+7=11$
$\therefore$ the point is $(1,11)$