Question:

The equation of a hyperbola with eccentricity $\frac{3}{2}$ and foci at $(\pm 2, 0)$ is determined as follows: Step 1: Identify the parameters from the given information. The foci of the hyperbola are at $(\pm c, 0)$. Given foci at $(\pm 2, 0)$, we have $c = 2$. The eccentricity is given as $e = \frac{3}{2}$. Step 2: Determine the semi-major axis $a$. The eccentricity of a hyperbola is defined by the relation $e = \frac{c}{a}$. Substituting the known values of $e$ and $c$: $$ \frac{3}{2} = \frac{2}{a} $$ Solving for $a$: $$ 3a = 4 \implies a = \frac{4}{3} $$ Therefore, $a^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$. Step 3: Determine the semi-minor axis $b$. For a hyperbola, the relationship between $a$, $b$, and $c$ is $c^2 = a^2 + b^2$. This can be rearranged to find $b^2$: $$ b^2 = c^2 - a^2 $$ Substituting the values $c^2 = 2^2 = 4$ and $a^2 = \frac{16}{9}$: $$ b^2 = 4 - \frac{16}{9} $$ $$ b^2 = \frac{36}{9} - \frac{16}{9} $$ $$ b^2 = \frac{20}{9} $$ Step 4: Write the equation of the hyperbola. The standard equation of a hyperbola centered at the origin with foci on the x-axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Substituting the calculated values of $a^2 = \frac{16}{9}$ and $b^2 = \frac{20}{9}$: $$ \frac{x^2}{\frac{16}{9}} - \frac{y^2}{\frac{20}{9}} = 1 $$ This equation can be rewritten by multiplying the numerator and denominator of each term by 9: $$ \frac{9x^2}{16} - \frac{9y^2}{20} = 1 $$