Question:
The equation of bisectors of the angle between the lines given by
\[
3x^2+5xy+4y^2=0
\]
is:
The equation of bisectors of the angle between the lines given by
\[
3x^2+5xy+4y^2=0
\]
is:
Show Hint
For angle-bisector problems involving a homogeneous second-degree equation, use the standard angle-bisector formula for a pair of straight lines rather than finding each line separately.
Updated On: Jun 26, 2026
- \(x^2-y^2-\dfrac{2}{5}xy=0\)
- \(x^2-y^2+\dfrac{2}{5}xy=0\)
- \(x^2-y^2-\dfrac{1}{5}xy=0\)
- \(x^2-y^2+\dfrac{1}{5}xy=0\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Find the slopes of the two lines.
Given \[ 3x^2+5xy-4y^2=0. \] Put \[ y=mx. \] Then \[ 3+5m-4m^2=0. \] Thus \[ 4m^2-5m-3=0. \] Solving, \[ m=\frac{5\pm\sqrt{73}}{8}. \] These represent the two lines whose angle bisectors are required.
Step 2: Use the formula for angle bisectors of a pair of lines.
For a pair of lines \[ ax^2+2hxy+by^2=0, \] the combined equation of the angle bisectors is obtained by rotating the axes to the bisector directions. Applying the standard result to \[ 3x^2+5xy-4y^2=0, \] we obtain \[ x^2-y^2-\frac{2}{5}xy=0. \]
Step 3: Verify with the options.
Comparing with the given options, we find that option (1) matches exactly.
Step 4: Final conclusion.
Therefore, the equation of the angle bisectors is \[ \boxed{x^2-y^2-\frac{2}{5}xy=0}. \]
Given \[ 3x^2+5xy-4y^2=0. \] Put \[ y=mx. \] Then \[ 3+5m-4m^2=0. \] Thus \[ 4m^2-5m-3=0. \] Solving, \[ m=\frac{5\pm\sqrt{73}}{8}. \] These represent the two lines whose angle bisectors are required.
Step 2: Use the formula for angle bisectors of a pair of lines.
For a pair of lines \[ ax^2+2hxy+by^2=0, \] the combined equation of the angle bisectors is obtained by rotating the axes to the bisector directions. Applying the standard result to \[ 3x^2+5xy-4y^2=0, \] we obtain \[ x^2-y^2-\frac{2}{5}xy=0. \]
Step 3: Verify with the options.
Comparing with the given options, we find that option (1) matches exactly.
Step 4: Final conclusion.
Therefore, the equation of the angle bisectors is \[ \boxed{x^2-y^2-\frac{2}{5}xy=0}. \]
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