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)

Given equation of hyperbola is
$16x^2-3y^2-32x-12y=44$
$\Rightarrow 16x^2+16-32x-3y^2-12-12y$
$=44+4$
$\Rightarrow 16(x-1{)}^2-3(y+2{)}^2=48$
$\Rightarrow \frac{(x-1{)}^2}{3}-\frac{(y+2{)}^2}{16}=1$
On comparing the equation with standard equation of hyperbola,
we get $a=\sqrt{3},b=4$
Now, length of transverse axis
$=2a=2\sqrt{3}$
and length of latusrectum
$=\frac{2b^2}{a}=\frac{2\times 16}{\sqrt{3}}=\frac{32}{\sqrt{3}}$
$\therefore$ Eccentricity $(e)=\sqrt{1+\frac{b^2}{a^2}}$
$=\sqrt{1+\frac{16}{3}}=\sqrt{\frac{19}{3}}$
Equation of directrix is $x=\pm \frac{a}{e}$
$\Rightarrow x-1=\pm \frac{\sqrt{3}\times \sqrt{3}}{\sqrt{19}}$
$\Rightarrow x=1\pm \frac{3}{\sqrt{19}}$