Straight Line and Pair of Straight Lines 1 Question 13
13. The tangent to the curve, $y=x e^{x^{2}}$ passing through the point $(1, e)$ also passes through the point
(2019 Main, 10 Jan II)
(a) $\frac{4}{3}, 2 e$
(b) $(3,6 e)$
(c) $(2,3 e)$
(d) $\frac{5}{3}, 2 e$
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Solution:
- Given equation of curve is $y=x e^{x^{2}}$
Note that $(1, e)$ lie on the curve, so the point of contact is $(1, e)$.
Now, slope of tangent, at point $(1, e)$, to the curve (i) is
$$ \begin{aligned} \left.\frac{d y}{d x}\right| _{(1, e)} & =x(2 x) e^{x^{2}}+e^{x^{2}} \\ & =2 e+e=3 e \end{aligned} $$
Now, equation of tangent is given by
$$ \begin{gathered} \left(y-y _1\right)=m\left(x-x _1\right) \\ y-e=3 e(x-1) \Rightarrow y=3 e x-2 e \end{gathered} $$
On checking all the options, the option $\frac{4}{3}, 2 e$ satisfy the equation of tangent.
