Question

The equations of the directrices of the hyperbola $16x^2-9y^2=-144$ are:

• A. $y=\pm \frac{8}{25}$
• B. $y=\pm \frac{16}{25}$
• C. $y=\pm \frac{8}{5}$
• D. $y=\pm \frac{16}{5}$

Answer 1

The correct option is D $y=\pm \frac{16}{5}$

$\mathrm{The}\mathrm{equation}16x^2-9y^2=-144\mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as}\frac{x^2}{9}-\frac{y^2}{16}=-1\mathrm{This}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1\mathrm{So},a^2=9,b^2=16\mathrm{or}a=3,b=4\mathrm{Also},\mathrm{eccentricity}\left(e\right)=\sqrt{\left(1+\frac{a^2}{b^2}\right)}=\sqrt{1+\frac{9}{16}}=\frac{5}{4}\mathrm{The}\mathrm{equations}\mathrm{of}\mathrm{the}\mathrm{directrices}\mathrm{are}\mathrm{given}\mathrm{by}:y=\pm \frac{b}{e}y=\pm \frac{4}{\left(\frac{5}{4}\right)}=\pm \frac{16}{5}$