If two circles touching both the axes are passing through (2,3) then length of their common chord is

Options

A. `sqrt(2)`
B. `2sqrt(2)`
C. `3sqrt(2)`
D. `4sqrt(2)`

Answer

To find the length of the common chord of two circles that touch both axes and pass through the point (2, 3), we can follow these steps:

### Step 1: Understand the Circle's Properties
Since both circles touch the x-axis and y-axis, their centers will be at (k, k) and (h, h) respectively, where k and h are the radii of the circles.

### Step 2: Write the Equations of the Circles
The equations of the circles can be written as:
1. Circle 1: \((x - k)^2 + (y - k)^2 = k^2\)
2. Circle 2: \((x - h)^2 + (y - h)^2 = h^2\)

### Step 3: Substitute the Point (2, 3)
Since both circles pass through the point (2, 3), we can substitute this point into both equations.

For Circle 1:
\[ (2 - k)^2 + (3 - k)^2 = k^2 \]
Expanding this:
\[ (2 - k)^2 + (3 - k)^2 = k^2 \]
\[ (4 - 4k + k^2) + (9 - 6k + k^2) = k^2 \]
\[ 13 - 10k + 2k^2 = k^2 \]
\[ k^2 - 10k + 13 = 0 \tag{1} \]

For Circle 2:
\[ (2 - h)^2 + (3 - h)^2 = h^2 \]
Expanding this:
\[ (2 - h)^2 + (3 - h)^2 = h^2 \]
\[ (4 - 4h + h^2) + (9 - 6h + h^2) = h^2 \]
\[ 13 - 10h + 2h^2 = h^2 \]
\[ h^2 - 10h + 13 = 0 \tag{2} \]

### Step 4: Solve the Quadratic Equations
We can solve equations (1) and (2) using the quadratic formula \(h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

For \(k\):
\[ k = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 13}}{2 \cdot 1} \]
\[ k = \frac{10 \pm \sqrt{100 - 52}}{2} \]
\[ k = \frac{10 \pm \sqrt{48}}{2} \]
\[ k = \frac{10 \pm 4\sqrt{3}}{2} \]
\[ k = 5 \pm 2\sqrt{3} \]

For \(h\):
\[ h = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 13}}{2 \cdot 1} \]
The calculation is identical to \(k\), yielding:
\[ h = 5 \pm 2\sqrt{3} \]

### Step 5: Find the Length of the Common Chord
The length of the common chord can be calculated using the formula:
\[ L = \sqrt{(k + h)^2 - (k - h)^2} \]
Substituting \(k\) and \(h\):
\[ L = \sqrt{( (5 + 2\sqrt{3}) + (5 - 2\sqrt{3}) )^2 - ( (5 + 2\sqrt{3}) - (5 - 2\sqrt{3}) )^2} \]
\[ L = \sqrt{(10)^2 - (4\sqrt{3})^2} \]
\[ L = \sqrt{100 - 48} = \sqrt{52} = 2\sqrt{13} \]

### Final Answer
The length of the common chord is \(2\sqrt{13}\).

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