Question

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

• A. Length of the transverse axis is $2\sqrt{3}$
• B. Length of each latusrectum is $\frac{32}{\sqrt{3}}$
• C. Eccentricity is $\sqrt{\frac{19}{3}}$
• 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 latusrectum is $\frac{32}{\sqrt{3}}$
C Eccentricity is $\sqrt{\frac{19}{3}}$

Given equation of hyperbola is
$16x^2-3y^2-33x-12y=44$
$\Rightarrow 16x^2+16-32x-3y^2-12-12y=44+4$
$\Rightarrow 16(x-1{)}^2-3(y+2{)}^2=48$
$\frac{(x-1{)}^2}{3}-\frac{(y+2{)}^2}{16}=1$
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 $\left(e\right)=\sqrt{1+\frac{b^2}{a^2}}$

$=\sqrt{1+\frac{16}{3}}=\sqrt{\frac{19}{3}}$

Equation of directrix is $x=\pm \frac{a}{e}$

$\Rightarrow x-1=\pm \frac{\sqrt{3}\times \sqrt{3}}{\sqrt{19}}$

$\Rightarrow x=1\pm \frac{3}{\sqrt{19}}$