HyperbolaHardBloom L1
Question
Let $F_{1},F_{2}$ are the foci of the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ and $F_{3},F_{4}$ are the foci of its conjugate hyperbola. If $e_{H}$ and $e_{C}$ are their eccentricities respectively then the statement which holds true is:
Options
A.Their equations of the asymptotes are different.
B.$e_{H} > e_{C}$
C.Area of the quadrilateral formed by their foci is 50 sq. units.
D.Their auxillary circles will have the same equation.
Solution
Sol. (c)
$$\begin{matrix} & \ 9 = 16\left( e_{H}^{2} - 1 \right) \Rightarrow e_{H} = \frac{5}{4} \\ & F_{1},F_{2} \equiv ( \pm 5,0), \\ & \ 16 = 19\left( e_{C}^{2} - 1 \right) \Rightarrow e_{C} = \frac{5}{3} \\ & F_{3},F_{4} \equiv (0, \pm 5) \\ & \text{~}\text{Area of}\text{~}F_{1}F_{3}F_{2}F_{4} = \frac{1}{2} \times 10 \times 10 = 50 \\ & \ \\ & \\ & \ \end{matrix}$$
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