Question

If $m$ be the slope of a tangent to the curve $e^{2y}=1+4x^2$, then

• A. $m<1$
• B. $\left|m\right|\le 1$
• C. $\left|m\right|>1$
• D. None of these

Answer 1

The correct option is B $\left|m\right|\le 1$

We have, $e^{2y}=1+4x^2\Rightarrow e^{2y}.2\frac{dy}{dx}=8x$

$\Rightarrow \frac{dy}{dx}=\frac{4x}{e^{2y}}=\frac{4x}{1+4x^2}.$

$\therefore$ Slope of tangent $=m=\frac{4x}{1+4x^2}$

$\Rightarrow \left|m\right|=\frac{4\left|x\right|}{1+4{\left|x\right|}^2}\le 1$

$[\because {(1-2\left|x\right|)}^2\ge 0\Rightarrow 1+4{\left|x\right|}^2-4\left|x\right|\ge 0\Rightarrow \frac{4\left|x\right|}{1+4{\left|x\right|}^2}\le 1]$