Question

Which of the statement in the following is false in respect of the conic $x^2-3y^2-4x-6y-11=0$

• A. Length of the latus rectum is $4/\sqrt{3}$
• B. Asymptotes, intersect at right angles
• C. The eccentricity of the conic is $2/\sqrt{3}$
• D. None of these

Answer 1

Answer: (B)

Asymptotes, intersect at right angles

$x^2-4x-3y^2-6y-11=0$

$\therefore (x-2{)}^2-4-3(y-1{)}^2+3-11=0$

$\therefore (x-2{)}^2-3(y-1{)}^2=12$

$\therefore \frac{(x-2{)}^2}{12}-\frac{(y-1{)}^2}{4}=1$

This is a hyperbola of the standard form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a=\sqrt{12},b=2$

Eccentricity $=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{4}{12}}=\frac{2}{\sqrt{3}}$

Latus rectum $=\frac{2b^2}{a}=\frac{2\times 4}{\sqrt{12}}=\frac{4}{\sqrt{3}}$

Angle between asymptotes $=2{\tan }^{-1}\left(\frac{b}{a}\right)=2{\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)={60}^o$

Hence, statement in option B is false.