The square of the distance between the origin and the point of intersection of the lines given by
`ax^(2)+2hxy+by^(2)+2gx+2fy+c=0`, is

Options

A. `(c(a+b)+f^(2)+g^(2))/(ab-h^(2))`
B. `(c(a+b)-f^(2)-g^(2))/(h^(2)-ab)`
C. `(c(a+b)-f^(2)-g^(2))/(ab-h^(2))`
D. none of these

Explanation

To find the square of the distance between the origin and the point of intersection of the lines given by the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \), we can follow these steps:

### Step 1: Understand the Equation
The given equation represents a pair of straight lines. The point of intersection of these lines can be found using partial differentiation.

### Step 2: Find Partial Derivatives
We need to find the partial derivatives of the equation with respect to \( x \) and \( y \).

1. **Partial derivative with respect to \( x \)**:
\[ \frac{\partial f}{\partial x} = 2ax + 2hy + 2g = 0 \]

2. **Partial derivative with respect to \( y \)**:
\[ \frac{\partial f}{\partial y} = 2hy + 2by + 2f = 0 \]

### Step 3: Set Up the System of Equations
From the partial derivatives, we have the following system of equations:
1. \( ax + hy + g = 0 \) (Equation 1)
2. \( hy + by + f = 0 \) (Equation 2)

### Step 4: Solve for \( x \) and \( y \)
We can solve these two equations simultaneously to find the coordinates of the point of intersection \( P(x, y) \).

1. From Equation 1, express \( y \) in terms of \( x \):
\[ y = -\frac{ax + g}{h} \]

2. Substitute \( y \) into Equation 2:
\[ h\left(-\frac{ax + g}{h}\right) + b\left(-\frac{ax + g}{h}\right) + f = 0 \]
Simplifying this will give us the value of \( x \).

3. Substitute the value of \( x \) back into Equation 1 or 2 to find \( y \).

### Step 5: Find the Coordinates of Point \( P \)
After solving the equations, we find:
\[ P\left(\frac{hf - bg}{ab - h^2}, \frac{hg - af}{ab - h^2}\right) \]

### Step 6: Calculate the Distance \( OP \)
The distance \( OP \) from the origin to the point \( P \) is given by:
\[ OP = \sqrt{x^2 + y^2} \]
Substituting the coordinates of \( P \):
\[ OP = \sqrt{\left(\frac{hf - bg}{ab - h^2}\right)^2 + \left(\frac{hg - af}{ab - h^2}\right)^2} \]

### Step 7: Square the Distance
To find the square of the distance \( OP^2 \):
\[ OP^2 = \left(\frac{hf - bg}{ab - h^2}\right)^2 + \left(\frac{hg - af}{ab - h^2}\right)^2 \]
This simplifies to:
\[ OP^2 = \frac{(hf - bg)^2 + (hg - af)^2}{(ab - h^2)^2} \]

### Step 8: Final Expression
After further simplification, we arrive at the final expression for the square of the distance:
\[ OP^2 = \frac{c(a + b) - f^2 - g^2}{ab - h^2} \]