HyperbolaHardBloom L3
Question
The differential equation $\frac{dx}{dy} = \frac{3y}{2x}$ represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity:
Options
A.$\sqrt{\frac{3}{5}}$
B.$\sqrt{\frac{5}{3}}$
C.$\sqrt{\frac{2}{5}}$
D.$\sqrt{\frac{5}{2}}$
Solution
Sol. (b, d)
$$\begin{matrix} & \ \int_{}^{}\ 2xdx = \int_{}^{}\ 3ydy \\ \Rightarrow \ & x^{2} = \frac{3y^{2}}{2} + c \\ \frac{x^{2}}{c} - \frac{y^{2}}{(2c/3)} = 1 & \\ \text{~}\text{If}\text{~}c < 0,c + \frac{2c}{3} = & \frac{2c}{3}e^{2} \\ \Rightarrow \ & \ e = \sqrt{\frac{5}{2}} \\ & \text{~}\text{If}\text{~}c > 0,c + \frac{2c}{3} = ce^{2} \\ \Rightarrow \ & e = \sqrt{\frac{5}{3}} \end{matrix}$$
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