Question

The combined equation of the bisectors of the angles between the lines joining the origin to the points of intersection of the curve \( x^2 + y^2 + xy + x + 3y + 1 = 0 \) and the line \( x + y + 2 = 0 \) is:

• A. \( x^2 + 4xy - y^2 = 0 \)
• B. \( x^2 - 4xy + y^2 = 0 \)
• C. \( 2x^2 - 3xy + y^2 = 0 \)
• D. \( x^2 + 2xy - 3y^2 = 0 \)

Hint:

In problems involving angle bisectors, always start by finding the points of intersection and then use the formula for the angle bisector equation.

Answer 1

The Correct Option is A

We are given the curve \( x^2 + y^2 + xy + x + 3y + 1 = 0 \) and the line \( x + y + 2 = 0 \). To find the combined equation of the bisectors of the angles formed by the lines joining the origin to the points of intersection of the curve and the line, we first find the points of intersection. Substitute \( y =
-x
- 2 \) into the equation of the curve and solve for \( x \). Once the points of intersection are found, we can use the standard formula for the combined equation of angle bisectors for two lines. After solving the equations, we arrive at the equation \( x^2 + 4xy
- y^2 = 0 \), which represents the combined equation of the bisectors.