Solve this following
Question:

The tangent to the curve, $y=x^{x^{2}}$ passing through the point (1,e) also passes through the point :

1. $\left(\frac{4}{3}, 2 \mathrm{e}\right)$

2. $(2,3 \mathrm{e})$

3. $\left(\frac{5}{3}, 2 \mathrm{e}\right)$

4. $(3,6 \mathrm{e})$

Correct Option: 1

Solution:

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

$\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{(1, e)}=\left.\left(\mathrm{e} \cdot \mathrm{e}^{\mathrm{x}^{2}} \cdot 2 \mathrm{x}+\mathrm{e}^{\mathrm{x}^{2}}\right)\right|_{(1, \mathrm{e})}=2 \cdot \mathrm{e}+\mathrm{e}=3 \mathrm{e}$

$\mathrm{T}: \mathrm{y}-\mathrm{e}=3 \mathrm{e}(\mathrm{x}-1)$

$y=3 e x-3 e+e$

$y=(3 e) x-2 e$

$\left(\frac{4}{3}, 2 e\right)$ lies on it

Option (1)