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If the pairs of straigth lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 be such that each pair bisects the angle between the other pair, then
  • a)
    p = -q
  • b)
    pq = 1
  • c)
    pq = -1
  • d)
    p = q
Correct answer is option 'C'. Can you explain this answer?
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If the pairs of straigth lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 =...
Explanation:


To find the relationship between the coefficients p and q, we need to determine the angle bisectors of the given pair of straight lines.

Angle bisectors of the first pair of lines:

The equation of the first pair of lines is x^2 - 2pxy - y^2 = 0.
To find the angle bisectors, we can rewrite the equation in the form (x - y)(x + y) - 2pxy = 0.
Let A = x - y and B = x + y.
Substituting these values into the equation, we get AB - 2p(A^2 - B^2)/4 = 0.
Simplifying further, we have AB - p(A^2 - B^2)/2 = 0.

The angle bisectors of this pair of lines are given by the equation AB - p(A^2 - B^2)/2 = 0.

Angle bisectors of the second pair of lines:

The equation of the second pair of lines is x^2 - 2qxy - y^2 = 0.
Similarly, rewriting the equation, we get AB - q(A^2 - B^2)/2 = 0.

The angle bisectors of this pair of lines are given by the equation AB - q(A^2 - B^2)/2 = 0.

Relationship between the angle bisectors:

Since each pair bisects the angle between the other pair, the angle bisectors of the first pair of lines coincide with the angle bisectors of the second pair of lines.

Comparing the equations of the angle bisectors, we have AB - p(A^2 - B^2)/2 = AB - q(A^2 - B^2)/2.

Canceling out the common terms, we get -p(A^2 - B^2)/2 = -q(A^2 - B^2)/2.

Dividing both sides by (A^2 - B^2)/2, we obtain -p = -q.

Therefore, the correct answer is option C: pq = -1.
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If the pairs of straigth lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 be such that each pair bisects the angle between the other pair, thena)p = -qb)pq = 1c)pq = -1d)p = qCorrect answer is option 'C'. Can you explain this answer?
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