Eduprobe
Question:
The locus of the point of intersection of the lines, √2x−y+4√2k=0 and √2kx+ky−√2=0 (k is any non-zero real parameter), is?
A hyperbola with length of its transverse axis 8√2
An ellipse with length of its major axis 8√2
A hyperbola whose eccentricity is √3
An ellipse whose eccentricity is √3
Solution:
Given lines are : √2x−y+4√2k=0 ⇒ √2x+4√2k=y (i) and √2kx+ky−√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 ⇒(y/4√2)²−(x/4)²=1 ⇒(y/4√2)²−(x/4)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.