Question:
The separate equations of the lines represented by $4x^2 - y^2 + 2x + y = 0$ are
The separate equations of the lines represented by $4x^2 - y^2 + 2x + y = 0$ are
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Homogeneous quadratic equations often represent a pair of straight lines.
Updated On: Feb 18, 2026
- $2x - 2y + 1 = 0,\ x + 2y = 0$
- $2x - y + 1 = 0,\ 2x + y = 0$
- $2x - y + 1 = 0,\ 2x - y = 0$
- $2x - y = 0,\ 2x + y + 1 = 0$
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The Correct Option is B
Solution and Explanation
Step 1: Rearranging the given equation.
\[ 4x^2 - y^2 + 2x + y = 0 \] \[ (2x)^2 - y^2 + 2x + y = 0 \]
Step 2: Grouping terms.
\[ (4x^2 + 2x) - (y^2 - y) = 0 \] \[ 2x(2x + 1) - y(y - 1) = 0 \]
Step 3: Writing as a product of linear factors.
\[ (2x - y + 1)(2x + y) = 0 \]
Step 4: Conclusion.
The separate equations of the lines are \[ 2x - y + 1 = 0 \quad \text{and} \quad 2x + y = 0 \]
\[ 4x^2 - y^2 + 2x + y = 0 \] \[ (2x)^2 - y^2 + 2x + y = 0 \]
Step 2: Grouping terms.
\[ (4x^2 + 2x) - (y^2 - y) = 0 \] \[ 2x(2x + 1) - y(y - 1) = 0 \]
Step 3: Writing as a product of linear factors.
\[ (2x - y + 1)(2x + y) = 0 \]
Step 4: Conclusion.
The separate equations of the lines are \[ 2x - y + 1 = 0 \quad \text{and} \quad 2x + y = 0 \]
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