Question

If a circle $S(x,y)=0$ touches at the point $(2,3)$ of the line $x+y=5$ and $S(1,2)=0$, then radius of such circle

• A. $2$ units
• B. $4$ units
• C. $\frac{1}{2}$ units
• D. $\frac{1}{\sqrt{2}}$ units

Answer 1

The correct option is D $\frac{1}{\sqrt{2}}$ units

Equation of given line is $x+y=5$

Slope of given line = $-1$

As this line is tangential to given circle at $(2,3)$, so a line perpendicular to given line passing through $(2,3)$ will also pass through center of given circle.

As product of slope of two mutually perpendicular lines =$-1$

So, slope of line perpendicular to given line =$1$

Equation of a line having slope =$1$ and passing through $(2,3)$ is $x-y=-1$

This line passes through $(1,2)$ and also the circle passes through $(1,2)$

So $(1,2)$ and $(2,3)$ are two diametrically opposite points on the circle.

Hence, radius = $\frac{\sqrt{(2-1{)}^2+(3-2{)}^2}}{2}$

$\Rightarrow$ radius = $\frac{1}{\sqrt{2}}$