Question

The area (in sq. units) enclosed between the curve $x=\frac{1 - t^{2}}{1 + t^{2}},y=\frac{2 t}{1 + t^{2}},\forall t\in R$ and the line $y=x+1$ above the line is

• A. $\frac{\pi }{4}$
• B. $\frac{1}{2}$
• C. $\frac{3 \pi }{4}+\frac{1}{2}$
• D. $\frac{\pi }{4}-\frac{1}{2}$

Answer 1

Answer: (D)

Let, $t=tan \theta $
$\Rightarrow x=cos 2\theta ,y=sin⁡2\theta $
$\Rightarrow x^{2}+y^{2}=1$

Area of the shaded region is the difference of $\frac{1}{4}$ of the circle and area of the triangle according to the figure.
$=\frac{\pi }{4}-\frac{1}{2}\times 1\times 1=\frac{\pi }{4}-\frac{1}{2}$ sq. units