If a circle whose centre is (2, 3) touches the line 4x + 3y − 7 = 0, then the radius of the circle is:

The correct answer is Option 1: 2 unit.

To find the radius of the circle, we can use the geometric property that the perpendicular distance from the centre of a circle to any line tangent to it (or touching it) is equal to its radius.

Step 1: Identify the given values.

Centre of the circle, $(x_1,y_1)=(2,3)$
Equation of the touching line: $4x+3y-7=0$

Here, comparing the line equation with the standard form $Ax+By+C=0$, we have:
$A=4$, $B=3$, and $C=\mathrm{-7}$.

Step 2: Apply the perpendicular distance formula.

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

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

Step 3: Calculate the radius.

Substitute the values into the formula:

$r=\frac{\left|4\right(2)+3(3)-7|}{\sqrt{4^2+3^2}}$

$r=\frac{|8+9-7|}{\sqrt{16+9}}$

$r=\frac{\left|10\right|}{\sqrt{25}}$

$r=\frac{10}{5}=2$

Thus, the radius of the circle is 2 unit.