If a chord of the circle `x^(2)+y^(2)=8` makes equal intercepts of length a on the coordinate axes, then `|a| lt `

Options

A. 2
B. 4
C. `sqrt(2)`
D. 8

Answer

To solve the problem, we need to find the condition for the length of the intercepts made by a chord of the circle \(x^2 + y^2 = 8\) on the coordinate axes.

### Step-by-Step Solution:

1. **Identify the Circle's Properties**:
The equation of the circle is given as \(x^2 + y^2 = 8\). This is a circle centered at the origin (0, 0) with a radius \(r = \sqrt{8} = 2\sqrt{2}\).

**Hint**: Remember that the radius is derived from the equation of the circle in the form \(x^2 + y^2 = r^2\).

2. **Equation of the Chord**:
Since the chord makes equal intercepts of length \(a\) on the coordinate axes, the equation of the chord can be expressed as:
\[ x + y = a \quad \text{and} \quad x - y = a \]
or in a more general form:
\[ x + y = k \quad \text{where } k = \pm a \]

**Hint**: Equal intercepts mean that the x-intercept and y-intercept are the same, leading to the form of the line.

3. **Finding the Perpendicular Distance**:
The distance \(d\) from the center of the circle (0, 0) to the line \(x + y = a\) can be calculated using the formula for the distance from a point to a line:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 1\), \(B = 1\), \(C = -a\), and the point is (0, 0):
\[ d = \frac{|0 + 0 - a|}{\sqrt{1^2 + 1^2}} = \frac{|a|}{\sqrt{2}} \]

**Hint**: The distance formula is crucial for determining the relationship between the chord and the circle.

4. **Condition for Intersection**:
For the chord to intersect the circle at two distinct points, the distance \(d\) must be less than the radius \(r\):
\[ \frac{|a|}{\sqrt{2}} < 2\sqrt{2} \]

**Hint**: Always compare the distance to the radius to ensure the chord intersects the circle.

5. **Solving the Inequality**:
To simplify the inequality:
\[ |a| < 2\sqrt{2} \cdot \sqrt{2} = 4 \]
Thus, we conclude:
\[ |a| < 4 \]

**Hint**: Simplifying the inequality helps in finding the final condition for \(a\).

### Final Answer:
The condition for the length of the intercepts made by the chord on the coordinate axes is:
\[ |a| < 4 \]