Question

The locus of the point of intersection of the lines \[\sqrt{3}x - y - 4\sqrt{3}\lambda = 0 \text { and } \sqrt{3}\lambda  + \lambda - 4\sqrt{3} = 0\]  is a hyperbola of eccentricity

Options

• 1

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Answer 1

2

The equations of lines 

\[\sqrt{3}x - y - 4\sqrt{3}\lambda = 0 \text { and } \sqrt{3}\lambda + \lambda - 4\sqrt{3} = 0\] can be rewritten as \[\sqrt{3}x - y = 4\sqrt{3}\lambda \text { and } \sqrt{3}\lambda + \lambda = 4\sqrt{3}\]  respectively.

Multiplying the equations: 

\[3\lambda x^2 - \lambda y^2 = 48\lambda\]

\[ \Rightarrow \frac{3\lambda x^2}{48\lambda} - \frac{\lambda y^2}{48\lambda} = 1\]

\[ \Rightarrow \frac{x^2}{16} - \frac{y^2}{48} = 1\]

This is the standard equation of a hyperbola, where  \[a^2 = 16 \text { and }b^2 = 48\].

\[\text { Eccentricity }, e = \sqrt{\frac{a^2 + b^2}{a^2}}\]

\[ \Rightarrow e = \sqrt{\frac{16 + 48}{16}}\]

\[ \Rightarrow e = \frac{8}{4}\]

\[ \Rightarrow e = 2\]