The lengths of intercepts made by any circle on the coordinate axes are equal if the centre lies on the line represented by

Options

A. `x ^(2) - y ^(2)=0`
B. `x - y =1`
C. `x + y=1`
D. none of these

Answer

To solve the problem, we need to find the condition under which the lengths of the intercepts made by any circle on the coordinate axes are equal, given that the center of the circle lies on a specific line.

### Step-by-Step Solution:

1. **Understanding the Circle and Intercepts**:
- Let the center of the circle be at point \( (h, k) \) and the radius be \( r \).
- The equation of the circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \).

2. **Finding Intercepts on the Axes**:
- The intercepts on the x-axis occur when \( y = 0 \):
\[ (x - h)^2 + (0 - k)^2 = r^2 \implies (x - h)^2 + k^2 = r^2. \]
Solving for \( x \) gives:
\[ x - h = \pm \sqrt{r^2 - k^2} \implies x = h \pm \sqrt{r^2 - k^2}. \]
Thus, the x-intercepts are \( h + \sqrt{r^2 - k^2} \) and \( h - \sqrt{r^2 - k^2} \).

- The intercepts on the y-axis occur when \( x = 0 \):
\[ (0 - h)^2 + (y - k)^2 = r^2 \implies h^2 + (y - k)^2 = r^2. \]
Solving for \( y \) gives:
\[ y - k = \pm \sqrt{r^2 - h^2} \implies y = k \pm \sqrt{r^2 - h^2}. \]
Thus, the y-intercepts are \( k + \sqrt{r^2 - h^2} \) and \( k - \sqrt{r^2 - h^2} \).

3. **Setting the Lengths of Intercepts Equal**:
- The lengths of the x-intercepts are:
\[ 2\sqrt{r^2 - k^2}. \]
- The lengths of the y-intercepts are:
\[ 2\sqrt{r^2 - h^2}. \]
- For the lengths to be equal, we set:
\[ \sqrt{r^2 - k^2} = \sqrt{r^2 - h^2}. \]

4. **Squaring Both Sides**:
- Squaring both sides gives:
\[ r^2 - k^2 = r^2 - h^2. \]
- This simplifies to:
\[ k^2 = h^2. \]

5. **Finding the Locus of the Center**:
- The equation \( k^2 = h^2 \) can be rewritten as:
\[ k - h = 0 \quad \text{or} \quad k + h = 0. \]
- This means:
\[ k = h \quad \text{or} \quad k = -h. \]
- In terms of coordinates, this represents the lines:
\[ y = x \quad \text{and} \quad y = -x. \]

6. **Final Combined Equation**:
- The combined equation of these two lines can be expressed as:
\[ (y - x)(y + x) = 0 \implies y^2 - x^2 = 0. \]

### Conclusion:
The lengths of intercepts made by any circle on the coordinate axes are equal if the center lies on the lines represented by \( y = x \) and \( y = -x \).