Q94 / 160 · WBJEE 2015 · Mathematics

Let $y=e^{x^{2}}$ and $y=e^{x^{2}} \sin x$ be two given curves. Then, angle between the tangents to the curves at any point of their intersection

  • Correct answer
    A
    0
  • B
    $\pi$
  • C
    $\frac{\pi}{2}$
  • D
    $\frac{\pi}{4}$

Solution

For intersecting points, $e^{x^{2}}=e^{x^{2}} \sin x$ $\Rightarrow \quad e^{x^{2}}(\sin x-1)=0$ $\Rightarrow \quad e^{x^{2}}=0 \quad$ or $\sin x=1$ But $\quad e^{x^{2}} \neq 0 \Rightarrow \sin x=1$ $\Rightarrow$ $$ x=\frac{\pi}{2} $$ Now. $$ y=e^{x^{2}} $$ $\therefore \quad \frac{d y}{d x}=e^{x^{2}} \cdot 2 x=2 x e^{x^{2}}$ Also, $y=e^{x^{2}} \sin x$ $\frac{d y}{d x}=e^{x^{2}} \cdot 2 x \sin x+e^{x^{2}} \cos x=2 x e^{x^{2}}$ $[\because \sin x=1, \cos x=0]$ Since, both the curves has equal slope. Hence, angle between the tangents at intersecting point is $0 .$

Hence, the correct answer is option: 'A'.

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