Question

If the radius of one circle is twice the radius of another circle whose centers are$(2,0)$ and $(1,2)$ respectively are cutting orthogonally, then the radius of the second circle is.

Answer 1

Step $1$: Compute the distance between the centers of the circle.

Given: Centers are$(2,0)$ and $(1,2)$.

Let's plot the figure.

Since the distance between two points is given by:

$D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Distance between the centers is

$\begin{array}{rcl}{C}_1{C}_2& =& \sqrt{{(2-1)}^2+{(0-2)}^2}\\ & \Rightarrow & C_1{C}_2=\sqrt{1+4}\\ & \Rightarrow & C_1{C}_2=\sqrt{5}\end{array}$

Step $2$: Compute the radius of the circle.

Given: Circles are cutting orthogonally.

So, the angle between the circles is $90°$.

Applying Pythagoras theorem:

$\begin{array}{rcl}{r_1}^2+{r_2}^2& =& {\left(C_1{C}_2\right)}^2\\ & \Rightarrow & {r_1}^2+{\left(2r_1\right)}^2={\left(\sqrt{5}\right)}^2\\ & \Rightarrow & 5{r_1}^2=5\\ & \Rightarrow & r_1=1\\ r_2& =& 2r_1=2\end{array}$ [ radius of one circle is twice the radius of another circle ]

Hence, the radius of the first circle is $1$ unit and the radius of the second circle is $2$ units.