The locus of the point of intersection of the lines 2x−y+42k=0 and 2kx+ky−42=0 (k is any non-zero real parameter) is:
Solutions · 1
Official solution
From the first equation, y=2x+42k; substituting into the second:
\[2\sqrt{2}kx+4\sqrt{2}(k^2-1)=0\ \Longrightarrow\ x=\frac{2(1-k^2)}k,\qquad y=\frac{2\sqrt{2}(1+k^2)}k.\]
Eliminating k:
\[\frac{y^2}{(4\sqrt{2})^2}-\frac{x^2}{4^2}=1.\]
Transverse axis length =2(42)=82.