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Question

The locus of the point of intersection of the lines, 2xy+42k=0 and 2kx+ky42=0 (k is any non-zero real parameter), is?

A
A hyperbola with length of its transverse axis 82
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B
An ellipse with length of its major axis 82
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C
An ellipse whose eccentricity is 13
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D
A hyperbola whose eccentricity is 3
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Solution

The correct option is A A hyperbola with length of its transverse axis 82
Given lines are :
2xy+42k=0

2x+42k=y ..... (i) and

2kx+ky42=0 ..... (ii)

We have from the equations of the lines:

Substituting (i) in (ii),

22kx+42(k21)=0

x=2(1k2)k,y=22(1+k2)k

(y42)2(x4)2=1

(y42)2(x4)2=1

Locus of transverse axis

=232

=2×42

=82

Thus, the locus is a hyperbola with length of its transverse axis equal to 82.
So option A is the correct answer.

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