QJEEM18Ma534f: JEE Main 2018 Mathematics - Class 11 JEEM18Ma534f Solution: The locus of the point of intersection of the lines, and (k is…

Question

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The locus of the point of intersection of the lines,

\sqrt 2 x - y + 4\sqrt 2 k = 0

and

\sqrt 2 k\,x + k\,y - 4\sqrt 2 = 0

(k is any non-zero real parameter), is :

Accepted Answer

Here, lines are :

\sqrt 2 x

-

y + 4

\sqrt 2 k

= 0

\Rightarrow

\,\,\,

\sqrt 2 x + 4\sqrt 2 k = y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....

(i)

and

\sqrt 2 kx + ky - 4\sqrt 2 = 0\,\,\,\,\,...\left( {ii} \right)

Put the value of y from (i) in (ii) we get;

\Rightarrow

2

\sqrt 2

kx + 4

\sqrt 2

(k²

-

1) = 0

\Rightarrow

x =

{{2\left( {1 - {k^2}} \right)} \over k}

, y =

{{2\sqrt 2 \left( {1 + {k^2}} \right)} \over k}

\therefore\,\,\,

{\left( {{y \over {4\sqrt 2 }}} \right)^2} - {\left( {{x \over 4}} \right)^2} = 1

\therefore\,\,\,

length of transverse axis

2a = 2

\times

4

{\sqrt 2 }

= 8

{\sqrt 2 }

Hence, the locus is a hyperbola with length of its transverse axis equal to 8

{\sqrt 2 }

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