If (4, 0) and (-4, 0) are the foci of an ellipse and the semiminor axis is 3, then the ellipse passes thrugh which one of the following points ?

Options

A. (2, 0)
B. (0, 5)
C. (0, 0)
D. (5, 0)

Explanation

To solve the problem step by step, we need to find the equation of the ellipse given its foci and semi-minor axis, and then determine which point lies on this ellipse.

### Step 1: Identify the foci and semi-minor axis
The foci of the ellipse are given as (4, 0) and (-4, 0). The semi-minor axis is given as 3.

### Step 2: Determine the value of 'c'
The distance from the center of the ellipse to each focus is denoted as 'c'. Since the foci are at (4, 0) and (-4, 0), the center of the ellipse is at (0, 0). Therefore, we have:
- c = 4

### Step 3: Determine the value of 'b'
The semi-minor axis 'b' is given as 3:
- b = 3

### Step 4: Use the relationship between a, b, and c
For an ellipse, the relationship between the semi-major axis 'a', semi-minor axis 'b', and the distance to the foci 'c' is given by:
\[ c^2 = a^2 - b^2 \]

Substituting the known values:
\[ 4^2 = a^2 - 3^2 \]
\[ 16 = a^2 - 9 \]
\[ a^2 = 16 + 9 \]
\[ a^2 = 25 \]
Thus,
\[ a = 5 \]

### Step 5: Write the standard equation of the ellipse
Since the major axis is along the x-axis, the standard equation of the ellipse is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values of a and b:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]

### Step 6: Check which point satisfies the equation
We need to check which of the given points satisfies the equation of the ellipse. Let's check the point (5, 0):
\[ \frac{5^2}{25} + \frac{0^2}{9} = \frac{25}{25} + 0 = 1 \]

Since the left-hand side equals 1, the point (5, 0) lies on the ellipse.

### Conclusion
The point through which the ellipse passes is (5, 0).

---