NIMCET 2025 — Mathematics PYQ

To find the area of the circle, we first need to determine its radius.

Step 1: Identify the Center of the Circle

Since the circle lies in the first quadrant and touches both the coordinate axes (the $x$-axis and the $y$-axis), its center $C$ must be at equal distances from both axes.

If the radius of the circle is $r$ (r > 0), then the coordinates of the center are:

$$C=(r,r)$$

Step 2: Use the Tangency Condition for the Given Line

The circle also touches the straight line:

$$x-y-2=0$$

The perpendicular distance from the center $C(r,r)$ to this tangent line must be equal to the radius $r$ of the circle.

The formula for the perpendicular distance from a point $(x_1,y_1)$ to a line $Ax+By+C=0$ is:

$$d=\frac{|A{x}_1+B{y}_1+C|}{\sqrt{A^2+B^2}}$$

Substituting the center $(r,r)$ and the line coefficients $A=1$, $B=-1$, and $C=-2$:

$$r=\frac{|1(r)-1(r)-2|}{\sqrt{1^2+(-1{)}^2}}$$

Simplify the expression inside the absolute value and the denominator:

$$r=\frac{|-2|}{\sqrt{1+1}}$$

$$r=\frac{2}{\sqrt{2}}$$

$$r=\sqrt{2}$$

Step 3: Calculate the Area of the Circle

The formula for the area of a circle is:

$$\text{Area}=\pi r^2$$

Substituting the value of $r=\sqrt{2}$:

$$\text{Area}=\pi (\sqrt{2}{)}^2=2\pi \text{ sq. units}$$

Since $2\pi$ can be rewritten to match typical entrance exam options, let's verify standard representations. Expanding Option B gives $\pi (4+2+4\sqrt{2})=\pi (6+4\sqrt{2})$. However, our exact computed radius value yields an area of $2\pi$.

Correct Answer

The exact area of the circle is:

$$2\pi \text{ sq. units}$$