Question

Which of the following is true for the hyperbola $9x^2-16y^2-18x+32y-151=0$?

• A. The length of the transverse axes is $4$
• B. Length of latus rectum is $9$
• C. Equation of directrix is $x=\frac{21}{5}$ and $x=-\frac{11}{5}$
• D. None of these

Answer 1

The correct option is C Equation of directrix is $x=\frac{21}{5}$ and $x=-\frac{11}{5}$

$9x^2-16y^2-18x+32y-151=0$
$9(x^2-2x)-16(y^2-2y)=151$
$9(x^2-2x+1)-16(y^2-2y+1)=151-7=144$
$9(x-1{)}^2-16(y-1{)}^2=144$
$\frac{(x-1{)}^2}{16}-\frac{(y-1{)}^2}{9}=1$
$\Rightarrow a^2=16,b^2=9$
$\therefore e=\sqrt{1+\frac{b^2}{a^2}}=\frac{5}{4}$
$\therefore$ The length of the transverse axis is $=2a=8$
Length of latus rectum is $=2\frac{b^2}{a}=\frac{9}{2}$
Equation of directrix is, $x=1\pm \frac{a}{e}=1\pm \frac{16}{5}=\frac{21}{5}$ or $-\frac{11}{5}$
Hence, option 'C' is correct.