If (4, 0)and (-4, 0) are the foci of ellipse and the semi-minor axis is 3, then the ellipse passes through which one of the following points?

Options

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

Answer

To solve the problem, we need to find the equation of the ellipse given its foci and semi-minor axis. Let's go through the steps systematically.

### Step 1: Identify the foci and semi-minor axis
The foci of the ellipse are given as (4, 0) and (-4, 0). This means the coordinates of the foci can be represented as \( (c, 0) \) and \( (-c, 0) \), where \( c = 4 \).

### Step 2: Determine the semi-major axis (a)
The semi-minor axis (b) is given as 3. We know the relationship between the semi-major axis (a), 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:
\[ 4^2 = a^2 - 3^2 \]
\[ 16 = a^2 - 9 \]
\[ a^2 = 16 + 9 = 25 \]
\[ a = 5 \]

### Step 3: Write the equation of the ellipse
The standard form of the equation of an ellipse centered at the origin with horizontal major axis 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 4: Check which point lies on the ellipse
We need to check which of the given points satisfies the equation of the ellipse. Let's check the point (5, 0):
\[ \frac{5^2}{25} + \frac{0^2}{9} = \frac{25}{25} + 0 = 1 \]
This satisfies the equation, so (5, 0) lies on the ellipse.

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

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