Question

The separate equations of the lines represented by $4x^2 - y^2 + 2x + y = 0$ are

• A. $2x - 2y + 1 = 0,\ x + 2y = 0$
• B. $2x - y + 1 = 0,\ 2x + y = 0$
• C. $2x - y + 1 = 0,\ 2x - y = 0$
• D. $2x - y = 0,\ 2x + y + 1 = 0$

Hint:

Homogeneous quadratic equations often represent a pair of straight lines.

Answer 1

The Correct Option is B

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 \[ 2x - y + 1 = 0 \quad \text{and} \quad 2x + y = 0 \]