If the quadratic equation `ax^2+bx+c=0` has `-2` as one of its roots then `ax + by + c = 0` represents

Options

A. A family of concurrent lines
B. A family of parallel lines
C. A single line
D. A line perpendicular to x - axis

Answer

To solve the problem, we need to analyze the given quadratic equation and the implications of having one of its roots as -2.

### Step-by-Step Solution:

1. **Understanding the Quadratic Equation**:
The quadratic equation is given as:
\[ ax^2 + bx + c = 0 \]
It is stated that -2 is one of its roots.

2. **Using the Root in the Equation**:
If -2 is a root, it must satisfy the equation. Therefore, we substitute \(x = -2\) into the equation:
\[ a(-2)^2 + b(-2) + c = 0 \]
This simplifies to:
\[ 4a - 2b + c = 0 \]

3. **Rearranging the Equation**:
We can rearrange this equation to express \(c\) in terms of \(a\) and \(b\):
\[ c = -4a + 2b \]

4. **Substituting \(c\) into the Line Equation**:
Now, we look at the line equation given:
\[ ax + by + c = 0 \]
Substituting the value of \(c\) we found:
\[ ax + by - 4a + 2b = 0 \]
Rearranging gives:
\[ ax + by = 4a - 2b \]

5. **Analyzing the Line Equation**:
The equation \(ax + by = 4a - 2b\) represents a line in the xy-plane. The coefficients of \(x\) and \(y\) can change based on the values of \(a\) and \(b\), but the right-hand side will always ensure that the line passes through a specific point.

6. **Finding the Point of Intersection**:
To find the specific point where these lines intersect, we can set \(x = 4\) and \(y = -2\). This means all lines represented by different values of \(a\) and \(b\) will pass through the point (4, -2).

7. **Conclusion**:
Since all lines represented by the equation \(ax + by + c = 0\) intersect at the point (4, -2), we conclude that this represents a family of concurrent lines.

### Final Answer:
The equation \(ax + by + c = 0\) represents a **family of concurrent lines**.

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