Question

The radius of the circle passing through the point $P(6,2)$, two of whose diameters are $x+y=6$ and $x+2y=4$ is

• A. $10$
• B. $2\sqrt{5}$
• C. $6$
• D. $4$

Answer 1

The correct option is B

$2\sqrt{5}$

Solve for the radius of the circle passing through the point $P(6,2)$

The point of intersection of the diameters is the center of the circle.

$$\begin{gathered} x+2y=4...\left(i\right) \\ x+y=6...\left(ii\right) \end{gathered}$$

$y=-2$

Substituting $y=-2$in $\left(ii\right)$

$$\begin{gathered} x-2=6 \\ \Rightarrow x=8 \end{gathered}$$

So, the center is $(8,-2)$

The circle passes through $(6,2)$

Radius $=\sqrt{{\left(8-6\right)}^2+{(-2-2)}^2}=\sqrt{20}$

$=2\sqrt{5}$

Hence, option(B) is the correct answer.