The pair of lines joining origin to the points of intersection of, the two curves `ax^2+2hxy + by^2+2gx = 0` and `a^'x^2 +2h^'xy + b^'y^2 + 2g^'x = 0` will be at right angles, if

Answer

To determine the condition under which the pair of lines joining the origin to the points of intersection of the two curves \( ax^2 + 2hxy + by^2 + 2gx = 0 \) and \( a'x^2 + 2h'xy + b'y^2 + 2g'x = 0 \) are at right angles, we can follow these steps:

### Step 1: Write the equations of the curves
The given equations are:
1. \( ax^2 + 2hxy + by^2 + 2gx = 0 \)
2. \( a'x^2 + 2h'xy + b'y^2 + 2g'x = 0 \)

### Step 2: Rearrange the equations
We can rearrange both equations to express them in a standard form:
1. \( ax^2 + 2hxy + by^2 = -2gx \)
2. \( a'x^2 + 2h'xy + b'y^2 = -2g'x \)

### Step 3: Set up the condition for intersection
The points of intersection of these two curves can be found by equating the left-hand sides of both equations:
\[ ax^2 + 2hxy + by^2 + 2gx = a'x^2 + 2h'xy + b'y^2 + 2g'x \]
This simplifies to:
\[ (a - a')x^2 + (2h - 2h')xy + (b - b')y^2 + (2g - 2g')x = 0 \]

### Step 4: Coefficients of the equation
From the above equation, we can identify the coefficients:
- Coefficient of \( x^2 \): \( a - a' \)
- Coefficient of \( xy \): \( 2(h - h') \)
- Coefficient of \( y^2 \): \( b - b' \)
- Coefficient of \( x \): \( 2(g - g') \)

### Step 5: Condition for the lines to be at right angles
For the pair of lines joining the origin to the points of intersection to be at right angles, the sum of the coefficients of \( x^2 \) and \( y^2 \) must equal zero:
\[ (a - a') + (b - b') = 0 \]
This implies:
\[ a + b = a' + b' \]

### Step 6: Conclusion
Thus, the condition for the pair of lines joining the origin to the points of intersection of the two curves to be at right angles is:
\[ gA' + B' = g'A + B \]