If `C_1` (1, 3) and `C_2`(4, 3)are the centres of two circles whose radical axis is y-axis. If the radius of the 1st circle is `2sqrt2` units, then the radius of the second circle is

Options

A. `sqrt23`
B. 3
C. 4
D. `2sqrt2`

Answer

To solve the problem step by step, we need to find the radius of the second circle given the centers and the radius of the first circle.

### Step 1: Identify the centers and radius of the circles
We are given:
- Center of Circle 1, \( C_1 = (1, 3) \)
- Center of Circle 2, \( C_2 = (4, 3) \)
- Radius of Circle 1, \( R_1 = 2\sqrt{2} \)

### Step 2: Write the equations of the circles
The general equation of a circle with center \((a, b)\) and radius \(r\) is given by:
\[ (x - a)^2 + (y - b)^2 = r^2 \]

For Circle 1:
\[ (x - 1)^2 + (y - 3)^2 = (2\sqrt{2})^2 \]
Calculating \( (2\sqrt{2})^2 \):
\[ (2\sqrt{2})^2 = 4 \times 2 = 8 \]
So, the equation for Circle 1 becomes:
\[ (x - 1)^2 + (y - 3)^2 = 8 \]

For Circle 2:
Let the radius of Circle 2 be \( R_2 \). The equation for Circle 2 is:
\[ (x - 4)^2 + (y - 3)^2 = R_2^2 \]

### Step 3: Find the radical axis
The radical axis of two circles can be found by subtracting the equations of the two circles. The radical axis is given to be the y-axis, which means it is represented by \( x = 0 \).

Subtracting the equations:
\[ [(x - 1)^2 + (y - 3)^2 - 8] - [(x - 4)^2 + (y - 3)^2 - R_2^2] = 0 \]

Expanding both equations:
1. For Circle 1:
\[ (x^2 - 2x + 1) + (y^2 - 6y + 9) - 8 = 0 \implies x^2 + y^2 - 2x - 6y + 2 = 0 \]
2. For Circle 2:
\[ (x^2 - 8x + 16) + (y^2 - 6y + 9) - R_2^2 = 0 \implies x^2 + y^2 - 8x - 6y + (25 - R_2^2) = 0 \]

### Step 4: Set up the equation for the radical axis
Now we can set up the equation for the radical axis:
\[ (x^2 + y^2 - 2x - 6y + 2) - (x^2 + y^2 - 8x - 6y + (25 - R_2^2)) = 0 \]
This simplifies to:
\[ -2x + 8x + 2 - 25 + R_2^2 = 0 \]
\[ 6x + R_2^2 - 23 = 0 \]

### Step 5: Substitute \( x = 0 \)
Since the radical axis is the y-axis, we substitute \( x = 0 \):
\[ 6(0) + R_2^2 - 23 = 0 \]
This simplifies to:
\[ R_2^2 - 23 = 0 \implies R_2^2 = 23 \]

### Step 6: Find \( R_2 \)
Taking the square root of both sides:
\[ R_2 = \sqrt{23} \]

### Final Answer
The radius of the second circle is \( \sqrt{23} \).

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