Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# The Locus of the Point of Intersection of the Lines √ 3 X − Y − 4 √ 3 λ = 0 and √ 3 λ + λ − 4 √ 3 = 0 is a Hyperbola of Eccentricity - Mathematics

MCQ

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

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#### Solution

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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  $a^2 = 16 \text { and }b^2 = 48$.

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

$\Rightarrow e = \sqrt{\frac{16 + 48}{16}}$

$\Rightarrow e = \frac{8}{4}$

$\Rightarrow e = 2$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 27 Hyperbola
Q 20 | Page 20