If two distinct chords, drawn from the point (p, q) on the circle `x^(2)+y^(2) =px+qy` (where pq `ne` ~ 0) are bisected by the x-axis, then

Options

A. `p^(2)=q^(2)`
B. `p^(2)=8q^(2)`
C. `p^(2) lt 8q^(2)`
D. `p^(2) gt 8q^(2)`

Answer

To solve the problem, we need to analyze the given conditions and derive the necessary equations step by step.

### Step-by-Step Solution:

1. **Understanding the Circle Equation**:
The equation of the circle is given as:
\[ x^2 + y^2 = px + qy \]
Rearranging gives:
\[ x^2 - px + y^2 - qy = 0 \]

2. **Identifying the Point (p, q)**:
The point (p, q) lies on the circle, which means it satisfies the circle's equation.

3. **Chords Bisected by the X-axis**:
Let the endpoints of the chords be A(x1, y1) and B(x2, y2). Since these chords are bisected by the x-axis, the midpoints of these chords will have a y-coordinate of 0. Therefore, we can denote the midpoints as:
\[ O(h, 0) \]
where \( h \) is the x-coordinate of the midpoint.

4. **Using the Midpoint Formula**:
The midpoint O of the chord AB can be expressed as:
\[ h = \frac{x1 + x2}{2}, \quad 0 = \frac{y1 + y2}{2} \]
From the second equation, we have:
\[ y1 + y2 = 0 \implies y2 = -y1 \]

5. **Substituting into the Circle Equation**:
Since points A and B lie on the circle, we can substitute these coordinates into the circle's equation:
\[ x1^2 + y1^2 = px1 + qy1 \]
\[ x2^2 + y2^2 = px2 + qy2 \]
Substituting \( y2 = -y1 \):
\[ x2^2 + (-y1)^2 = px2 - qy1 \]

6. **Expressing x2 in terms of h**:
From the midpoint relation, we have:
\[ x1 = 2h - p \quad \text{and} \quad x2 = 2h - p \]

7. **Setting Up the Quadratic Equation**:
Substitute \( x1 \) and \( x2 \) into the circle's equation to form a quadratic in terms of h. After simplification, we will have:
\[ 4h^2 + (terms \, in \, h) + (constant \, terms) = 0 \]

8. **Finding the Discriminant**:
For the quadratic equation to have real solutions, the discriminant must be greater than zero:
\[ b^2 - 4ac > 0 \]
where \( a, b, c \) are coefficients from the quadratic equation.

9. **Solving the Inequality**:
After calculating the discriminant, we will derive a condition involving p and q:
\[ 36p^2 - 32p^2 - 32q^2 > 0 \implies 4p^2 > 32q^2 \implies p^2 > 8q^2 \]

### Conclusion:
Thus, the condition that must be satisfied is:
\[ p^2 > 8q^2 \]