If coordinates of focus (±5, 0) & vertices (±13, 0), then equation of ellipse will be

Question

Accepted Answer

The correct answer is Option 4: $\frac{x^{2}}{169}+\frac{y^{2}}{144}=1$.

### Explanation:

To determine the equation of the ellipse, we need to understand the properties of an ellipse and how its equation is derived from its geometric features.

1. **Vertices and Foci**: 
   - The given vertices are $(\pm13, 0)$, which means the major axis is along the x-axis, and the length of the major axis is $2a = 26$. Therefore, $a = 13$.
   - The given foci are $(\pm5, 0)$, indicating that the distance from the center to each focus is $c = 5$.

2. **Relationship between $a$, $b$, and $c$**:
   - For an ellipse, the relationship between the semi-major axis $a$, the semi-minor axis $b$, and the distance to the foci $c$ is given by the equation $c^2 = a^2 - b^2$.
   - Substituting the known values: $5^2 = 13^2 - b^2$.
   - This simplifies to $25 = 169 - b^2$.
   - Solving for $b^2$, we get $b^2 = 144$.

3. **Equation of the Ellipse**:
   - The standard form of the equation of an ellipse with a horizontal major axis is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
   - Substituting the values of $a^2 = 169$ and $b^2 = 144$, the equation becomes $\frac{x^2}{169} + \frac{y^2}{144} = 1$.

### Why Option 4 is Correct:
- Option 4 matches the derived equation $\frac{x^2}{169} + \frac{y^2}{144} = 1$, which is consistent with the given vertices and foci.

### Why Other Options are Incorrect:
1. **Option 1: $\frac{x^{2}}{169}-\frac{y^{2}}{144}=1$**:
   - This represents a hyperbola, not an ellipse, because of the subtraction sign. The problem specifies an ellipse.

2. **Option 2: $\frac{x^{2}}{144}+\frac{y^{2}}{169}=1$**:
   - This equation suggests that the major axis is along the y-axis, which contradicts the given vertices $(\pm13, 0)$.

3. **Option 3: $\frac{x^{2}}{144}-\frac{y^{2}}{169}=1$**:
   - Similar to Option 1, this represents a hyperbola due to the subtraction sign, which is incorrect for an ellipse.

In conclusion, Option 4 is the only choice that correctly represents the equation of an ellipse with the given vertices and foci.