If each of the pairs of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy- y2 =0 bisects the angles between the other pair, thena)pq=1b)pq=-1c)p2=-q2d)p2q2 = -1Correct answer is option 'B'. Can you explain this answer?

Question

Answer

The equation of the bisectors of the angles between the two straight lines given by ax²+2hxy+by²=0 is given by

Now we are given two pairs of straight lines, namely,
x²- 2 pxy - y² = 0 ...(I)
and x²- 2qxy-y² = 0 ...(II)
It is given that each of I or II is the bisectors of the angle between the other pair. Using (A) the equation of the bisectors of the angle between the straight lines represented by (I) is given by

Answer

Explanation:

Given equations:
The given pair of equations are x^2 - 2pxy - y^2 = 0 and x^2 - 2qxy - y^2 = 0.

Angle bisectors:
To bisect the angles between the pair of lines, the lines themselves must be the angle bisectors of the other pair.

Angle bisector condition:
The condition for a line to be an angle bisector of the angle between two lines is given by the equation Ac^2 = B^2 - AB, where A, B, and C are the coefficients of x^2, xy, and y^2 terms respectively in the general equation of a straight line Ax^2 + 2Bxy + Cy^2 = 0.

Applying the condition:
By applying the angle bisector condition to the given pairs of lines, we get:
For x^2 - 2pxy - y^2 = 0:
(1)^2 = (-2p)^2 - 1*(-1) => 1 = 4p^2 + 1 => 4p^2 = 0 => p^2 = 1/4
For x^2 - 2qxy - y^2 = 0:
(1)^2 = (-2q)^2 - 1*(-1) => 1 = 4q^2 + 1 => 4q^2 = 0 => q^2 = 1/4

Comparing p^2 and q^2:
From the above calculations, we can see that p^2 = q^2 = 1/4. Therefore, pq = (1/2)(1/2) = 1/4.

Conclusion:
Hence, the correct answer is pq = 1, which is represented by option 'B'.

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