Q.Find the equation of the hyperbola with eccentricity $\dfrac{3}{2}$ and foci at $(\pm 2, 0)$.

The key idea is understanding the Standard Form of a Hyperbola and the relationships between its parameters ($a, b, c, e$).

1. The foci are given as $(\pm 2, 0)$. This indicates the hyperbola is centered at the origin $(0,0)$ and its transverse axis lies along the x-axis. From the foci, we have $c=2$.
2. The eccentricity is given as $e = \frac{3}{2}$. For a hyperbola, $e = \frac{c}{a}$. Substituting the values, we get $\frac{3}{2} = \frac{2}{a}$, which implies $3a = 4$, so $a = \frac{4}{3}$.
3. For a hyperbola, the relationship between $a, b, c$ is $c^2 = a^2 + b^2$. Substituting $c=2$ and $a=\frac{4}{3}$: $2^2 = \left(\frac{4}{3}\right)^2 + b^2$ $4 = \frac{16}{9} + b^2$ $b^2 = 4 - \frac{16}{9} = \frac{36 - 16}{9} = \frac{20}{9}$. …

To find the equation of a hyperbola, we need to determine its orientation (whether the transverse axis is horizontal or vertical) and the values of its key parameters, $a^2$ and $b^2$. The given information — the foci and eccentricity — directly helps us find these.

The standard form of a hyperbola centered at the origin depends on where its foci lie.

• If the foci are on the x-axis at $(\pm c, 0)$, the transverse axis is horizontal, and the equation is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
• If the foci are on the y-axis at $(0, \pm c)$, the transverse axis is vertical, and the equation is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.

The eccentricity $e$ of a hyperbola is defined as the ratio $e = \frac{c}{a}$, where $c$ is the distance from the center to a focus, and $a$ is the distance from the center to a vertex. For a hyperbola, $e > 1$.

The relationship between $a, b, c$ for a hyperbola is $c^2 = a^2 + b^2$. This can also be expressed as $b^2 = a^2(e^2 - 1)$.

Let's apply these concepts to the given problem.

1. Identify the type of hyperbola and its center: The foci are given as $(\pm 2, 0)$. Since the y-coordinate is zero, the foci lie on the x-axis. This means the transverse axis of the hyperbola is along the x-axis, and the hyperbola is centered at the origin $(0,0)$. Therefore, the standard form of its equation will be:

2. Determine the value of $c$:

   The foci are at $(\pm c, 0)$. Comparing this with the given foci $(\pm 2, 0)$, we find that $c = 2$.

3. Determine the value of $a$ using eccentricity:

   We are given the eccentricity $e = \frac{3}{2}$.

   The definition of eccentricity for a hyperbola is $e = \frac{c}{a}$.

   Substitute the known values of $e$ and $c$:

Now, solve for $a$:

Then, $a^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$.

4. Determine the value of $b^2$:

We can use the relationship $b^2 = a^2(e^2 - 1)$. This is often more direct when eccentricity is given.

Substitute the values of $a^2$ and $e$: …