Question

The area (in sq units) of the triangle formed by the normal at point $(1,0)$ on the curve $x = e^{\sin y}$ with coordinate axes is:

• A. $1$
• B. $\dfrac{1}{4}$
• C. $\dfrac{1}{2}$
• D. $\dfrac{3}{8}$

Hint:

Normal and Triangle Area:

• Tangent slope $\Rightarrow$ normal slope $= -1/m$
• Normal + axes intersection $\Rightarrow$ triangle
• Area = $\frac12 \cdot x_\textint \cdot y_\textint$

Answer 1

The Correct Option is C

Given: $x = e^{\sin y} \Rightarrow \frac{dx}{dy} = e^{\sin y} \cos y$ \[ \Rightarrow \frac{dy}{dx} = \frac{1}{e^{\sin y} \cos y} \Rightarrow \left. \frac{dy}{dx} \right|_{(1,0)} = 1 \Rightarrow \text{slope of normal } = -1 \] Equation of normal: \[ y - 0 = -1(x - 1) \Rightarrow x + y = 1 \] Intercepts: $(1,0)$ and $(0,1) \Rightarrow \text{Area} = \frac{1}{2}(1)(1) = \boxed{\frac{1}{2}}$