The length of the common chord of the circles $(x-a)^{2}+y^{2}=c^{2}$ and $x^{2}+(y-b)^{2}=c^{2}$ is equal to :

Question

Accepted Answer

To determine the length of the common chord of the given circles, we need to analyze the geometric configuration of the circles and apply the appropriate mathematical principles.

### Explanation for Option 2 as the Correct Answer:

The two circles given are:
1. Circle 1: $(x-a)^2 + y^2 = c^2$
2. Circle 2: $x^2 + (y-b)^2 = c^2$

Both circles have the same radius $c$, but different centers: $(a, 0)$ for Circle 1 and $(0, b)$ for Circle 2.

#### Step-by-step Analysis:

1. **Distance Between Centers:**
   The distance $d$ between the centers of the two circles is calculated using the distance formula:
   $$
   d = \sqrt{(a - 0)^2 + (0 - b)^2} = \sqrt{a^2 + b^2}
   $$

2. **Condition for Intersection:**
   For the circles to intersect, the distance between their centers must be less than or equal to the sum of their radii and greater than or equal to the absolute difference of their radii. Since both radii are equal to $c$, the condition simplifies to:
   $$
   |c - c| \leq \sqrt{a^2 + b^2} \leq c + c
   $$
   $$
   0 \leq \sqrt{a^2 + b^2} \leq 2c
   $$

3. **Length of the Common Chord:**
   The length $L$ of the common chord of two intersecting circles can be found using the formula:
   $$
   L = \sqrt{4c^2 - d^2}
   $$
   Substituting $d = \sqrt{a^2 + b^2}$ into the formula, we get:
   $$
   L = \sqrt{4c^2 - (a^2 + b^2)}
   $$

This matches Option 2: $\sqrt{4c^2 - a^2 - b^2}$.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $\sqrt{4c^2 + a^2 + b^2}$**
   - This option incorrectly adds $a^2 + b^2$ to $4c^2$. The correct formula involves subtracting $a^2 + b^2$ because the length of the chord depends on the difference between the squared radius terms and the squared distance between centers.

2. **Option 3: $\sqrt{c^2 + a^2 + b^2}$**
   - This option does not account for the factor of 4 in the $4c^2$ term, which is crucial for calculating the length of the common chord. It also incorrectly adds $a^2 + b^2$ instead of subtracting it.

3. **Option 4: $\sqrt{c^2 - a^2 - b^2}$**
   - This option uses $c^2$ instead of $4c^2$, which does not correctly represent the geometric relationship needed to find the length of the common chord.

In conclusion, Option 2 is the correct answer because it accurately reflects the geometric and algebraic principles involved in determining the length of the common chord of two intersecting circles.