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If the pairs of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 be such that each pair bisects the angle between the other pair, then pq is equal to
Correct answer is '-1'. Can you explain this answer?
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If the pairs of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 ...
To find the value of pq, we need to determine the angle bisectors of the given pairs of straight lines.
Given pair 1: x^2 - 2pxy - y^2 = 0
Given pair 2: x^2 - 2qxy - y^2 = 0
Let's analyze each pair separately to find their angle bisectors.
Angle Bisectors for Pair 1:
The equation of a line passing through the origin at an angle θ with the x-axis is given by:
tan(θ) = 2p / (1 - p^2)
To find the angle bisector of the given pair 1, we need to find the angle (θ1) for which this equation holds true.
Angle Bisectors for Pair 2:
Similarly, the equation of a line passing through the origin at an angle φ with the x-axis is given by:
tan(φ) = 2q / (1 - q^2)
To find the angle bisector of the given pair 2, we need to find the angle (φ1) for which this equation holds true.
Angle Bisectors for Pair 1 bisecting Pair 2:
Now, let's find the angle bisector of pair 1 that bisects the angle between pair 2. Let this angle be α.
The equation of the angle bisector of pair 1 that bisects the angle between pair 2 is given by:
tan(α) = (tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1))
Since this line bisects the angle between pair 1 and pair 2, it should also be the angle bisector for pair 2 bisecting pair 1.
Hence, we can equate α with the angle bisector of pair 2, which is φ1.
Solving the equation tan(α) = (tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1)) = tan(φ1), we can find the value of α.
Now, the angle bisector of pair 1 bisecting pair 2 is α, and the angle bisector of pair 2 bisecting pair 1 is φ1.
Since these angles are bisectors, they should be equal.
Therefore, α = φ1
Simplifying the equation, we get:
(tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1)) = tan(φ1)
Cross-multiplying and simplifying, we have:
tan(θ1) + tan(φ1) = tan(φ1) - tan(θ1) * tan(φ1)^2
Rearranging the terms, we get:
2 * tan(φ1) * tan(θ1) - tan(θ1) - tan(φ1) = 0
This equation should hold true for any value of θ1 and φ1.
Comparing the coefficients, we get:
2 * tan(φ1) * tan(θ1) = 1
tan(θ1) = 1 / (2 * tan(φ1))
Comparing this equation with the equation of the angle bisector for pair 1, we have:
Given pair 1: x^2 - 2pxy - y^2 = 0
Given pair 2: x^2 - 2qxy - y^2 = 0
Let's analyze each pair separately to find their angle bisectors.
Angle Bisectors for Pair 1:
The equation of a line passing through the origin at an angle θ with the x-axis is given by:
tan(θ) = 2p / (1 - p^2)
To find the angle bisector of the given pair 1, we need to find the angle (θ1) for which this equation holds true.
Angle Bisectors for Pair 2:
Similarly, the equation of a line passing through the origin at an angle φ with the x-axis is given by:
tan(φ) = 2q / (1 - q^2)
To find the angle bisector of the given pair 2, we need to find the angle (φ1) for which this equation holds true.
Angle Bisectors for Pair 1 bisecting Pair 2:
Now, let's find the angle bisector of pair 1 that bisects the angle between pair 2. Let this angle be α.
The equation of the angle bisector of pair 1 that bisects the angle between pair 2 is given by:
tan(α) = (tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1))
Since this line bisects the angle between pair 1 and pair 2, it should also be the angle bisector for pair 2 bisecting pair 1.
Hence, we can equate α with the angle bisector of pair 2, which is φ1.
Solving the equation tan(α) = (tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1)) = tan(φ1), we can find the value of α.
Now, the angle bisector of pair 1 bisecting pair 2 is α, and the angle bisector of pair 2 bisecting pair 1 is φ1.
Since these angles are bisectors, they should be equal.
Therefore, α = φ1
Simplifying the equation, we get:
(tan(θ1) + tan(φ1)) / (1 - tan(θ1) * tan(φ1)) = tan(φ1)
Cross-multiplying and simplifying, we have:
tan(θ1) + tan(φ1) = tan(φ1) - tan(θ1) * tan(φ1)^2
Rearranging the terms, we get:
2 * tan(φ1) * tan(θ1) - tan(θ1) - tan(φ1) = 0
This equation should hold true for any value of θ1 and φ1.
Comparing the coefficients, we get:
2 * tan(φ1) * tan(θ1) = 1
tan(θ1) = 1 / (2 * tan(φ1))
Comparing this equation with the equation of the angle bisector for pair 1, we have:
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Community Answer
If the pairs of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 ...
Given equations are
x2 - 2qxy - y2 = 0 ... (1)
x2 - 2pxy - y2 = 0 ... (2)
Joint equations of angle bisector of the lines (i) and (ii) are same.
qx2 + 2xy - qy2 = 0 ... (3)
Now, (2) and (3) are same.
Taking ratio of their coefficients.
⇒ pq = -1
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If the pairs of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 be such that each pair bisects the angle between the other pair, then pq is equal toCorrect answer is '-1'. Can you explain this answer?
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