Q.Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola $3x^2 - y^2 = 4$.

$3x^2-y^2=4 \Rightarrow \dfrac{x^2}{4/3}-\dfrac{y^2}{4}=1$. So $a^2=\dfrac43,\,b^2=4 \Rightarrow a=\dfrac{2\sqrt3}{3},\,b=2$.

$e^2=1+\dfrac{b^2}{a^2}=1+\dfrac{4}{4/3}=1+3=4 \Rightarrow e=2$.

• Transverse axis $=2a=\dfrac{4\sqrt3}{3}$; conjugate axis $=2b=4$.
• Foci $(\pm ae,0)=\left(\pm\dfrac{2\sqrt3}{3}\times2,0\right)=\left(\pm\dfrac{4\sqrt3}{3},0\right)$.
• Directrices $x=\pm\dfrac{a}{e}=\pm\dfrac{2\sqrt3/3}{2}=\pm\dfrac{\sqrt3}{3}$. …

Divide by the constant to get the fractional $a^2$, then carry the fraction careful …

Rounding $a^2=4/3$ or dropping the fraction, which corrupt …