QJEEM21Ma6bff: JEE Main 2021 Mathematics - Class 11 JEEM21Ma6bff Solution: The locus of the point of intersection of the lines and is a…

Question

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The locus of the point of intersection of the lines

\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0

and

\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 0

is a conic, whose eccentricity is _________.

Accepted Answer

\sqrt 3 kx + ky = 4\sqrt 3

........(1)

\sqrt 3 kx - ky = 4\sqrt 3 {k^2}

....... (2)

Adding equation (1) & (2)

2\sqrt 3 kx = 4\sqrt 3 ({k^2} + 1)

x = 2\left( {k + {1 \over k}} \right)

......... (3)

Substracting equation (1) & (2)

y = 2\sqrt 3 \left( {{1 \over k} - k} \right)

........(4)

\therefore

{{{x^2}} \over 4} - {{{y^2}} \over {12}} = 4

{{{x^2}} \over {16}} - {{{y^2}} \over {48}} = 1

(Hyperbola)

\therefore

{e^2} = 1 + {{48} \over {16}}

\Rightarrow

e = 2

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