Two straight lines having variable slopes m _ 1 and m _ 2 pass through the fixed points (a, 0) and (- a, 0) respectively. If m _ 1 m _ 2 = 2 , then the eccentricity of the locus of the point of intersection of the lines

Correct answer: B. 3 Let, the lines are y = m _ 1 (x - a) and y = m _ 2 (x + a) which intersects at the point (h, k) . ⇒ k = m _ 1 (h - a) & k = m _ 2 ( h + a) ⇒ k^ 2 = m _ 1 m _ 2 ( h^ 2 - a^ 2 ) ⇒ k^ 2 2 =h^ 2 - a^ 2 (given, m _ 1 m _ 2 = 2 ) ⇒ x^ 2 1 - y^ 2 2 =a^ 2 (Hyperbola) Eccentricity = 1 + (2)/(1) = 3

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Hyperbola

JEE Main · Mathematics · Hyperbola

Two straight lines having variable slopes m _ 1 and m _ 2 pass...

Two straight lines having variable slopes

{m}_{1}

and

{m}_{2}

pass through the fixed points \left(a, 0\right)

and \left(- a, 0\right)

respectively. If

{m}_{1}{m}_{2} = 2

, then the eccentricity of the locus of the point of intersection of the lines is

Options

A $\sqrt{2}$
B $\sqrt{3}$
C $2$
D $\sqrt{\dfrac{3}{2}}$

Correct answer

B. $\sqrt{3}$

Step-by-step solution

Let, the lines are

y = {m}_{1} \left(x - a\right)

and y = {m}_{2} \left(x + a\right)

<br/><br/>which intersects at the point \left(h, k\right) .

⇒ k = {m}_{1}\left(h - a\right)

k = {m}_{2} \left(

h + a\right)

<br/><br/>

⇒ k^{2} = {m}_{1}{m}_{2} \left(

h^{2} - a^{2}\right)

⇒ \dfrac{k^{2}}{2} =h^{2} - a^{2}

(given,

{m}_{1}{m}_{2} = 2

)

⇒ \dfrac{x^{2}}{1} - \dfrac{y^{2}}{2} =a^{2}

(Hyperbola)

Eccentricity

= \sqrt{1 + \dfrac{2}{1}} = \sqrt{3}

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