Question:
Suppose the pairs of straight lines \(x^2 - 2axy - y^2 = 0\) and \(x^2 - 2bxy - y^2 = 0\) are such that each pair bisects the angles between the other two. Then \(ab =\)
Suppose the pairs of straight lines \(x^2 - 2axy - y^2 = 0\) and \(x^2 - 2bxy - y^2 = 0\) are such that each pair bisects the angles between the other two. Then \(ab =\)
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For pairs of lines that are angle bisectors of each other, use the condition on slopes: product of slopes of lines in one pair = -1 of slopes in the other pair.
Updated On: Jul 18, 2026
- 1
- -1
- 2
- \(\frac{1}{2}\)
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The Correct Option is B
Solution and Explanation
Step 1: Condition for angle bisectors.
If two pairs of lines \(x^2 - 2 a xy - y^2 = 0\) and \(x^2 - 2 b xy - y^2 = 0\) bisect each other, then the product of slopes of the lines of one pair equals -1 of the other.
Step 2: Find slopes.
Slopes of first pair: \(m_1, m_2 = a \pm \sqrt{a^2 + 1}\)
Slopes of second pair: \(m_3, m_4 = b \pm \sqrt{b^2 + 1}\)
Step 3: Angle bisector condition.
Product of slopes condition: \(ab = -1\)
Step 4: Verification.
This satisfies the property that each pair bisects angles between the other pair.
Step 5: Final conclusion.
Hence, \[ \boxed{-1} \]
If two pairs of lines \(x^2 - 2 a xy - y^2 = 0\) and \(x^2 - 2 b xy - y^2 = 0\) bisect each other, then the product of slopes of the lines of one pair equals -1 of the other.
Step 2: Find slopes.
Slopes of first pair: \(m_1, m_2 = a \pm \sqrt{a^2 + 1}\)
Slopes of second pair: \(m_3, m_4 = b \pm \sqrt{b^2 + 1}\)
Step 3: Angle bisector condition.
Product of slopes condition: \(ab = -1\)
Step 4: Verification.
This satisfies the property that each pair bisects angles between the other pair.
Step 5: Final conclusion.
Hence, \[ \boxed{-1} \]
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