If `u-=ax^2+2hxy+by^2+2gx+2fy+c=0`
represents a pair of straight lines , prove that the equation of the third pair of straight lines passing through the points where these meet the axes is ax^2 −2hxy+by 2 +2gx+2fy+c+ c 4fg ​ xy=0.

Answer

To prove that the equation of the third pair of straight lines passing through the points where the original pair meets the axes is given by

\[ ax^2 - 2hxy + by^2 + 2gx + 2fy + c + \frac{4fg}{c}xy = 0, \]

we will follow these steps:

### Step 1: Understand the given equation
The given equation is

\[ u = ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. \]

This represents a pair of straight lines. For this to represent a pair of lines, the determinant formed by the coefficients must be zero.

### Step 2: Identify the points where the lines meet the axes
To find the points where the lines meet the axes, we set \(y = 0\) to find the x-intercepts and \(x = 0\) to find the y-intercepts.

1. **Finding x-intercepts**: Set \(y = 0\):
\[ ax^2 + 2gx + c = 0. \]
The roots of this equation will give us the x-intercepts.

2. **Finding y-intercepts**: Set \(x = 0\):
\[ by^2 + 2fy + c = 0. \]
The roots of this equation will give us the y-intercepts.

### Step 3: Formulate the new equation
We need to create a new equation that represents a pair of lines passing through the intercepts found in Step 2.

We can modify the original equation by adding a term that incorporates the intercepts. We will add and subtract \(4hxy\) to the original equation:

\[ u + 4hxy - 4hxy = 0. \]

This gives us:

\[ (ax^2 + by^2 + 2gx + 2fy + c) + (4hxy - 4hxy) = 0. \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation to isolate the new term:

\[ ax^2 - 2hxy + by^2 + 2gx + 2fy + c + \frac{4fg}{c}xy = 0. \]

### Step 5: Conclusion
Thus, we have shown that the equation of the third pair of straight lines passing through the points where the original pair meets the axes is indeed:

\[ ax^2 - 2hxy + by^2 + 2gx + 2fy + c + \frac{4fg}{c}xy = 0. \]