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 line on which the center of a circle lies such that the lengths of the intercepts made by the circle on the coordinate axes are equal.

### Step-by-Step Solution:

1. **Understanding the Circle and Intercepts**:
A circle can be represented in the standard form as \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. The intercepts on the x-axis and y-axis are the points where the circle intersects these axes.

2. **Finding Intercepts**:
- The x-intercepts occur when \(y = 0\):
\[ (x - h)^2 + (0 - k)^2 = r^2 \implies (x - h)^2 + k^2 = r^2 \]
This leads to:
\[ x - h = \pm \sqrt{r^2 - k^2} \implies x = h \pm \sqrt{r^2 - k^2} \]
Thus, the x-intercepts are \(h + \sqrt{r^2 - k^2}\) and \(h - \sqrt{r^2 - k^2}\).

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

3. **Setting the Intercepts Equal**:
For the lengths of the intercepts to be equal, we need:
\[ \text{Length of x-intercept} = \text{Length of y-intercept} \]
The lengths of the intercepts can be calculated as:
- Length of x-intercept = \(2\sqrt{r^2 - k^2}\)
- Length of y-intercept = \(2\sqrt{r^2 - h^2}\)

Setting these equal gives:
\[ \sqrt{r^2 - k^2} = \sqrt{r^2 - h^2} \]

4. **Squaring Both Sides**:
Squaring both sides, we have:
\[ r^2 - k^2 = r^2 - h^2 \]
Simplifying this, we find:
\[ k^2 = h^2 \]

5. **Conclusion**:
This implies that:
\[ k = \pm h \]
Therefore, the center of the circle \((h, k)\) lies on the lines represented by:
\[ y = x \quad \text{and} \quad y = -x \]

### Final Answer:
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 \(y = -x\).

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