Foci of the hyperbola are (-2, 0) and (2, 0). If its eccentricity is 2, then its equation will be:

Question

Accepted Answer

To determine the equation of the hyperbola with foci at (-2, 0) and (2, 0) and an eccentricity of 2, we need to follow these steps:

1. **Identify the Center and Orientation:**
   - The foci of the hyperbola are given as (-2, 0) and (2, 0). The midpoint of the line segment joining the foci is the center of the hyperbola. Thus, the center is at (0, 0).
   - Since the foci are on the x-axis, the transverse axis is horizontal.

2. **Determine the Values of 'a', 'b', and 'c':**
   - The distance between the center and each focus is 'c'. Here, c = 2.
   - The eccentricity (e) is given as 2. The relationship between the eccentricity, 'a', and 'c' for a hyperbola is $ e = \frac{c}{a} $.
   - Substituting the given eccentricity, $ 2 = \frac{2}{a} $, we find $ a = 1 $.

3. **Calculate 'b':**
   - For a hyperbola, the relationship between 'a', 'b', and 'c' is given by $ c^2 = a^2 + b^2 $.
   - Substituting the known values, $ 2^2 = 1^2 + b^2 $, which simplifies to $ 4 = 1 + b^2 $.
   - Solving for 'b', we get $ b^2 = 3 $.

4. **Formulate the Equation:**
   - The standard form of the equation of a hyperbola with a horizontal transverse axis is $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $.
   - Substituting the values of 'a' and 'b', we have $ \frac{x^2}{1^2} - \frac{y^2}{3} = 1 $, which simplifies to $ x^2 - \frac{y^2}{3} = 1 $.
   - Multiplying through by 3 to eliminate the fraction gives $ 3x^2 - y^2 = 3 $.

Thus, the correct equation of the hyperbola is $ 3x^2 - y^2 = 3 $, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1: $-3x^2 + y^2 = 3$**
  - This equation represents a hyperbola with a vertical transverse axis, which contradicts the given horizontal orientation of the foci.

- **Option 2: $x^2 - 3y^2 = 3$**
  - This equation suggests a hyperbola with a horizontal transverse axis, but the coefficients do not match the derived values for 'a' and 'b'. Specifically, the coefficient of $y^2$ should be negative and equal to 1, not 3.

- **Option 4: $-x^2 + 3y^2 = 3$**
  - This equation represents a hyperbola with a vertical transverse axis, which is inconsistent with the horizontal orientation indicated by the foci.

Therefore, Option 3 is the only equation that correctly represents the hyperbola with the given foci and eccentricity.