Question

Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. $49y^2-16x^2=784$.

Answer 1

Given Hyperbola: $49y^2-16x^2=784$

Dividing by 784 $\Rightarrow \frac{y^2}{16}-\frac{x^2}{49}=1$

As its a conjugate Hyperbola,

where $a=9,b=4$

Vertices : $(0,\pm b),(0,\pm 4)$

Eccentricity:e$=\sqrt{1+\frac{a^2}{b^2}}=\sqrt{1+\frac{49}{16}}=\sqrt{\frac{16+49}{16}}=\frac{1}{4}\sqrt{6}5$

Length of LR$=2\cdot \frac{a^2}{b}=2\cdot \frac{49}{4}=\frac{49}{2}$ unit length

Foci:$(0,\pm be):(0,\pm \sqrt{6}5)$