Question

Let $16x^2-3y^2-32x-12y=44$ represent a hyperbola. Then

• A. length of the transverse axis is $2\sqrt{3}$
  
• B. length of each latus rectum is $32/\sqrt{3}$
• C. eccentricity is $\sqrt{19/3}$
• D. equation of a directrix is $x=\frac{\sqrt{19}}{3}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A length of the transverse axis is $2\sqrt{3}$

B length of each latus rectum is $32/\sqrt{3}$
C eccentricity is $\sqrt{19/3}$

$16x^2-3y^2-32x-12y=44$
$\Rightarrow \left(\right(4x{)}^2-\mathrm{2.4.}\left(4x\right)+16-16)-(3y^2+2\sqrt{12}(\sqrt{3}y)+12-12)=44$
$\Rightarrow (4x-4{)}^2-(\sqrt{3}y+\sqrt{12}{)}^2=48$
$\Rightarrow 16(x-1{)}^2-3(y+2{)}^2=48$
$\Rightarrow \frac{(x-1{)}^2}{(\sqrt{3}{)}^2}-\frac{(y+2{)}^2}{(4{)}^2}=1$

$a=\sqrt{3},b=4$
$\therefore c=\sqrt{19}$

Length of the transverse axis: $2a=2\sqrt{3}.$
Length of latus rectum: $\frac{2b^2}{a}=\frac{32}{\sqrt{3}}$
Eccentricity: $\frac{c}{a}=\frac{\sqrt{19}}{\sqrt{3}}$
Equation of directrix:
$x=1+\frac{a}{e}=1+\frac{\sqrt{3}}{\sqrt{19}}$