Question

If the circles of same radius $a$ and centres at $(2,3)$ and $(5,6)$ cut orthogonally, then $a$ is equal to

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

As we know that if two circles cut each other orthogonally, then

${\left(C_1{C}_2\right)}^2={r_1}^2+{r_2}^2$

Therefore,

$C_1=(2,3)$

$C_2=(5,6)$

$r_1=r_2=a$

By using distance formula,

$C_1{C}_2=\sqrt{{(6-3)}^2+{(5-2)}^2}$

$\Rightarrow {\left(C_1{C}_2\right)}^2=3^2+3^2$

$\Rightarrow {\left(C_1{C}_2\right)}^2=9+9=18$

Therefore,

${\left(C_1{C}_2\right)}^2=a^2+a^2$

$18=2a^2$

$\Rightarrow a^2=\frac{18}{2}$

$\Rightarrow a=\sqrt{9}=3$