Find the area of the triangle formed by the lines represented by `ax^2+2hxy+by^2+2gx+2fy+c=0` and axis of x .

Answer

To find the area of the triangle formed by the lines represented by the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) and the x-axis, we can follow these steps:

### Step 1: Identify the lines
The given equation represents a pair of straight lines. To find these lines, we will set \( y = 0 \) in the equation, since the lines intersect the x-axis.

\[ ax^2 + 2gx + c = 0 \]

### Step 2: Find the roots of the equation
Now we need to find the roots of the quadratic equation \( ax^2 + 2gx + c = 0 \). The roots can be found using the quadratic formula:

\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Here, \( A = a \), \( B = 2g \), and \( C = c \).

\[ x = \frac{-2g \pm \sqrt{(2g)^2 - 4ac}}{2a} \]

\[ x = \frac{-2g \pm \sqrt{4g^2 - 4ac}}{2a} \]

\[ x = \frac{-g \pm \sqrt{g^2 - ac}}{a} \]

Let the roots be \( x_1 \) and \( x_2 \).

### Step 3: Calculate the area of the triangle
The area \( A \) of the triangle formed by the x-axis and the lines can be calculated using the formula for the area of a triangle:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

In this case, the base is the distance between the points \( (x_1, 0) \) and \( (x_2, 0) \), which is \( |x_2 - x_1| \), and the height is the y-coordinate of the intersection point of the lines with the y-axis (which is 0 here since they intersect the x-axis).

The distance between the roots \( x_1 \) and \( x_2 \) is given by:

\[ |x_2 - x_1| = \left| \frac{-g + \sqrt{g^2 - ac}}{a} - \frac{-g - \sqrt{g^2 - ac}}{a} \right| \]

Simplifying this gives:

\[ |x_2 - x_1| = \left| \frac{2\sqrt{g^2 - ac}}{a} \right| = \frac{2\sqrt{g^2 - ac}}{a} \]

Thus, the area \( A \) becomes:

\[ A = \frac{1}{2} \times \frac{2\sqrt{g^2 - ac}}{a} \times 0 = 0 \]

Since the height is 0, the area of the triangle formed by the lines and the x-axis is 0.

### Final Answer
The area of the triangle formed by the lines represented by \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) and the x-axis is:

\[ \text{Area} = 0 \]