Question

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

• A. m < 1
• B.
  
• C.
  
• D. None of these

Answer 1

The correct option is B

We have,
$e^{2y}=1+4x^2\Rightarrow e^{2y}.2\frac{dy}{dx}=8x\Rightarrow \frac{dy}{dx}=\frac{4x}{e^{2y}}=\frac{4x}{1+4x^2}\therefore Slopeoftangent=m=\frac{4x}{1+4x^2}\Rightarrow |m|=\frac{4|x|}{1+4|x{|}^2}\le 1[\begin{array}{c}\because (1-2)|x^2|\ge 0\Rightarrow 1+4|x{|}^2-4|x|\ge 0\\ \Rightarrow \frac{4|x|}{1+4|x{|}^2}\end{array}$

Hence (b) is the correct answer.