Question

The joint equation of bisectors of the angle between the lines represented by $3x^2+2xy-y^2=0$ is

• A. $x^2-4xy-y^2=0$
• B. $x^2+4xy-y^2=0$
• C. $x^2-4xy+y^2=0$
• D. $x^2+4xy+y^2=0$

Hint:

For a pair of straight lines through the origin, angle bisectors can be found using coefficient comparison.

Answer 1

The Correct Option is A

Step 1: Writing the given equation in standard form.
\[ 3x^2+2xy-y^2=0 \] This represents a pair of straight lines through the origin.
Step 2: Using the formula for angle bisectors.
For $ax^2+2hxy+by^2=0$, the joint equation of bisectors is \[ (ax^2-by^2)^2=4h^2x^2y^2 \]
Step 3: Substituting values.
Here $a=3$, $h=1$, $b=-1$. \[ (3x^2+y^2)^2=4x^2y^2 \] Taking square root and simplifying: \[ x^2-4xy-y^2=0 \]
Step 4: Conclusion.
The joint equation of the angle bisectors is $x^2-4xy-y^2=0$.