Question

If m be the slope of a tangent to the curve
${e}^{y} = 1 + {x}^{2}$
then
1) |m|>1
2) m<1
3) |m|<1
4) |m|<= 1​

Answer 1

Question :

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

1) |m|>1

2) m<1

3) |m|<1

4) |m|<= 1

Theory:

Let y=f(x) be a curve and let P(x,y) be a point on it . Then

$slope \: of \: tangent = \frac{dy}{dx}$

Solution :

we have to find slope of tangent to the given curve :

$e {}^{y} = 1 + x {}^{2}$

Differentiate the curve equation with respect to x

$\implies e {}^{y} \frac{dy}{dx} = 2x$

$\implies m = \frac{dy}{dx} = \frac{2x}{e {}^{y} } = \frac{2x}{1 + x {}^{2} }$

$\ |m| = \frac{2 |x| }{1 + |x {}^{2} | }$

$since \: x {}^{2} \geqslant 0$

$\implies \: x {}^{2} + 1 \geqslant 1$

$\implies \: \frac{1}{ x{}^{2} + 1 } \leqslant 1$

$\implies \: \frac{2x}{1 + x {}^{2} } \leqslant 1$

$\implies \: m \leqslant 1$

Therefore correct option d)