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Question

The locus of the point of intersection of the lines 3xy43k=0 and 3kx+ky43=0 for different values of k is the hyperbola _______

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Solution

Let point of intersection of lines is (α,β)

So, the point (α,β) lies on both the lines.

3xy43k=0

3αβ43k=0

k=3αβ43 (1)

And 3kx+ky43=0

3kα+kβ43=0

3α×3αβ43+β×3αβ4343=0 [From equation (1)]

3α23αβ+3αββ248=0

3α2β2=48

For the locus replace (α β) by (x,y)

3x2y2=48

Hence, the locus of point of intersection of lines 3xy43k=0 and 3kx+ky43=0 for the different value of k is the hyperbola 3x2y2=48.

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