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

$m < 1$

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$| M | \leq 1$

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| M | > 1

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$m = 2$

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Solution

The correct option is B
$| M | \leq 1$

We have,
$e^{2 y} = 1 + 4 x^{2} \Rightarrow e^{2 y} . 2 \frac{d y}{d x} = 8 x \Rightarrow \frac{d y}{d x} = \frac{4 x}{e^{2 y}} = \frac{4 x}{1 + 4 x^{2}} ∴ S l o p e o f t a n g e n t = m = \frac{4 x}{1 + 4 x^{2}} \Rightarrow | m | = \frac{4 | x |}{1 + 4 | x |^{2}} \leq 1 ⎡ ⎢ ⎣ \begin{matrix} ∵ (1 - 2) | x^{2} | \geq 0 \Rightarrow 1 + 4 | x |^{2} - 4 | x | \geq 0 \Rightarrow \frac{4 | x |}{1 + 4 | x |^{2}} \end{matrix}$
Hence (b) is the correct answer.

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

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

m

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