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Question

# The locus of the point of intersection of the lines, √2 x−y+4√2 k=0 and √2 kx+ky−4√2=0 (k is any non-zero real parameter), is

A
an ellipse whose eccentricity is 13
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B
a hyperbola whose eccentricity is 3
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C
a hyperbola having length of its transverse axis 82 units
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D
an ellipse having length of its major axis 82 units
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Solution

## The correct option is C a hyperbola having length of its transverse axis 8√2 units√2 x−y=−4√2 k .....(1) √2 x+y=4√2k .....(2) Multiplying both equation (√2x+y)(√2x−y−4√2)=4√2 ⇒2x2−y2=−32 ⇒y232−x216=1 Hence, it represents a hyperbola having transverse axis along y−axis ⇒e=√1+1632=√32 Also, length of transverse axis 2√32= 8√2 units

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