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# Find the locus of the point of intersection of the lines, (root 2)x-y+4(root2)k=0 and (root2)kx+ky-4(root 2)k=0 ( k is any non-zero real parameter) is?

Samyak Jain
333 Points
2 years ago
Equation of first line : $\dpi{80} \sqrt{2}$ x – y + 4$\dpi{80} \sqrt{2}$ k = 0    ….(1) and equation of second line : $\dpi{80} \sqrt{2}$ k x – k y + 4$\dpi{80} \sqrt{2}$ k = 0  i.e.
$\dpi{80} \sqrt{2}$ x + y – 4$\dpi{80} \sqrt{2}$ = 0      ….(2).
First equation represents parallel lines with constant slope ($\dpi{80} \sqrt{2}$) but varying intercept as k is parameter.
When these lines will intersect the second given line, the locus of point of intersection is nothing but the second line.
Hence required equation of locus is $\dpi{80} \sqrt{2}$ x + y – 4$\dpi{80} \sqrt{2}$ = 0.