The distance between the focii of the ellipse `x = 3 cos theta, y = 4 sin theta` is

Options

A. `sqrt(7)`
B. `sqrt(2)`
C. `2sqrt(7)`
D. `3sqrt(7)`

Answer

To find the distance between the foci of the ellipse given by the parametric equations \( x = 3 \cos \theta \) and \( y = 4 \sin \theta \), we can follow these steps:

### Step 1: Identify the semi-major and semi-minor axes
The parametric equations represent an ellipse in the form:
- \( x = a \cos \theta \)
- \( y = b \sin \theta \)

From the given equations:
- \( a = 3 \)
- \( b = 4 \)

Here, \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively. Since \( b > a \), we have:
- Semi-major axis \( b = 4 \)
- Semi-minor axis \( a = 3 \)

### Step 2: Calculate the distance between the foci
The distance between the foci of an ellipse is given by the formula:
\[ d = 2c \]
where \( c \) is the distance from the center to each focus, calculated using:
\[ c = \sqrt{b^2 - a^2} \]

### Step 3: Calculate \( b^2 \) and \( a^2 \)
Calculate \( b^2 \) and \( a^2 \):
- \( b^2 = 4^2 = 16 \)
- \( a^2 = 3^2 = 9 \)

### Step 4: Substitute into the formula for \( c \)
Now substitute these values into the formula for \( c \):
\[ c = \sqrt{b^2 - a^2} = \sqrt{16 - 9} = \sqrt{7} \]

### Step 5: Calculate the distance between the foci
Now, substitute \( c \) into the formula for the distance \( d \):
\[ d = 2c = 2\sqrt{7} \]

### Final Answer
The distance between the foci of the ellipse is:
\[ \boxed{2\sqrt{7}} \]

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