The lines joining the origin to the points of intersection of the curve
`ax^(2)+2hxy+by^(2)+2gx=0`
and `a_(1)x^(2)+2b_(1)xy+b_(1)y^(2)+2g_(1)x=0` are `_|_` then

Options

A. `(a+b)/(g_(1))=(a_(1)+b_(1))/g`
B. `(a+b)g_(1)=(a_(1)+b_(1))g`
C. `(a-b)g=(a_(1)-b_(1))g_(1)`
D. none of these

Answer

To solve the problem, we need to determine the condition under which the lines joining the origin to the points of intersection of the two given curves are perpendicular.

### Step-by-Step Solution:

1. **Identify the curves**: We have two curves represented by the equations:
- Curve 1: \( ax^2 + 2hxy + by^2 + 2gx = 0 \)
- Curve 2: \( a_1x^2 + 2b_1xy + b_1y^2 + 2g_1x = 0 \)

2. **Find the points of intersection**: To find the points of intersection of these curves, we can set them equal to each other. This will involve solving the two equations simultaneously.

3. **Formulate the determinant**: The lines joining the origin to the points of intersection can be represented in terms of their slopes. If the slopes of the lines from the origin to the points of intersection are \( m_1 \) and \( m_2 \), then the condition for these lines to be perpendicular is given by:
\[ m_1 \cdot m_2 = -1 \]

4. **Use the condition for perpendicularity**: The slopes can be derived from the coefficients of the equations. The condition for the lines to be perpendicular can be expressed in terms of the coefficients of the quadratic equations. Specifically, if we denote the coefficients of the first curve as \( A = a, B = 2h, C = b \) and the second curve as \( A_1 = a_1, B_1 = 2b_1, C_1 = b_1 \), then the condition for perpendicularity can be derived from the determinant of the coefficients.

5. **Set up the determinant**: The determinant condition for the two curves to intersect at right angles can be set up as follows:
\[ \begin{vmatrix} a & h & g \\ a_1 & b_1 & g_1 \\ 0 & 0 & 1 \end{vmatrix} = 0 \]
This determinant must equal zero for the lines to be perpendicular.

6. **Solve the determinant**: Expanding the determinant, we can derive the condition that relates the coefficients \( a, b, h, a_1, b_1, g, g_1 \).

7. **Final condition**: The final condition will give us a relationship between the coefficients of the two curves that must hold true for the lines from the origin to the points of intersection to be perpendicular.