Question

The area enclosed by the curves$\left|y+x\right|\le 1$, $\left|y-x\right|\le 1$ and $2x^2+2y^2=1$ is

• A. $\left(2+\frac{\mathrm{\pi }}{2}\right)\text{sq units}$
• B. $\left(2-\frac{\mathrm{\pi }}{2}\right)\text{sq units}$
• C. $\left(3+\frac{\mathrm{\pi }}{2}\right)\text{sq units}$
• D. $\left(3-\frac{\mathrm{\pi }}{4}\right)\text{sq units}$

Answer 1

The correct option is B

$\left(2-\frac{\mathrm{\pi }}{2}\right)\text{sq units}$

Explanation for correct answer Option(s)

Finding the area enclosed by the curves.

The given Equation of the curves are,

$\left|y+x\right|\le 1......\left(1\right)$

$\left|y-x\right|\le 1......\left(2\right)$

The Equation of a circle is

$2x^2+2y^2=1$

Rewrite the above Equation as,

$$\begin{gathered} \Rightarrow 2\left(x^2+y^2\right)=1 \\ \Rightarrow x^2+y^2=\frac{1}{2} \\ \Rightarrow x^2+y^2={\left(\frac{1}{\sqrt{2}}\right)}^2 \end{gathered}$$

Here the radius of a circle is $r=\frac{1}{\sqrt{2}}$ with center $\left(0,0\right)$

The graph is as shown:

The Equation $\left(1\right)$ and Equation $\left(2\right)$ from a square $ABCD$

$y+x=1$

Put $x=0$ we get

$$\begin{gathered} \Rightarrow y+0=1 \\ \Rightarrow y=1 \end{gathered}$$

Put $y=0$ we get

$$\begin{gathered} \Rightarrow 0+x=1 \\ \Rightarrow x=1 \end{gathered}$$

Hence, the points are $\left(0,1\right)\left(1,0\right)$

Then,

$y-x=1$

Put $x=0$ we get

$$\begin{gathered} \Rightarrow y-0=1 \\ \Rightarrow y=1 \end{gathered}$$

Put $y=0$ we get

$$\begin{gathered} \Rightarrow 0-x=1 \\ \Rightarrow x=-1 \end{gathered}$$

Hence, the points are $\left(0,1\right)\left(-1,0\right)$

From the graph $BC$ is equal to

$$\begin{gathered} BC=\sqrt{{\left(0-1\right)}^2+{\left(1-0\right)}^2} \\ BC=\sqrt{1+1} \\ BC=\sqrt{2} \end{gathered}$$

Also From the graph $DC$ is equal to

$$\begin{gathered} DC=\sqrt{{\left(0+1\right)}^2+{\left(1-0\right)}^2} \\ DC=\sqrt{1+1} \\ DC=\sqrt{2} \end{gathered}$$

Now area of a square $ABCD$ is Equal to

$$\begin{gathered} DC\times BC=\sqrt{2}\times \sqrt{2} \\ =2\text{sq units} \end{gathered}$$

Now the area of a shaded region is Equal to

$$\begin{gathered} A=\text{Area of square-Area of circle} \\ A=2-{\mathrm{\pi r}}^2 \\ \mathrm{A}=2-\mathrm{\pi }{\left(\frac{1}{\sqrt{2}}\right)}^2 \\ \mathrm{A}=2-\frac{\mathrm{\pi }}{2}\text{sq units} \end{gathered}$$

Therefore, the correct answer is Option B.