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Question

# The locus of the point of intersection of the lines, √2x−y+4√2k=0 and √2kx+ky−4√2=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 8√2Given lines are :√2x−y+4√2k=0⇒√2x+4√2k=y ..... (i) and√2kx+ky−4√2=0 ..... (ii)We have from the equations of the lines:Substituting (i) in (ii),⇒2√2kx+4√2(k2−1)=0⇒x=2(1−k2)k,y=2√2(1+k2)k ⇒(y4√2)2−(x4)2=1⇒(y4√2)2−(x4)2=1Locus of transverse axis=2√32=2×4√2=8√2Thus, the locus is a hyperbola with length of its transverse axis equal to 8√2. So option A is the correct answer.

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