"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 _________."

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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)

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

$\Rightarrow$ $e = 2$"

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 _________.

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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 _________.

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