An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cm are

Options

A. `6,2sqrt(5)`
B. `6,sqrt(5)`
C. `4,2sqrt(5)`
D. `6 + 2sqrt(5), 2sqrt(5)`

Answer

To solve the problem, we need to find the necessary length of the string and the distance between the pins for the ellipse with given axes.

### Step-by-Step Solution:

1. **Identify the semi-major and semi-minor axes**:
- The lengths of the axes are given as 6 cm (major axis) and 4 cm (minor axis).
- Therefore, the semi-major axis \( a \) is:
\[ a = \frac{6}{2} = 3 \text{ cm} \]
- And the semi-minor axis \( b \) is:
\[ b = \frac{4}{2} = 2 \text{ cm} \]

2. **Calculate the distance between the foci (pins)**:
- The distance between the foci \( c \) can be calculated using the formula:
\[ c = \sqrt{a^2 - b^2} \]
- First, calculate \( a^2 \) and \( b^2 \):
\[ a^2 = 3^2 = 9, \quad b^2 = 2^2 = 4 \]
- Now, substitute these values into the formula:
\[ c = \sqrt{9 - 4} = \sqrt{5} \]
- The distance between the two foci (pins) is:
\[ \text{Distance between pins} = 2c = 2\sqrt{5} \text{ cm} \]

3. **Calculate the necessary length of the string**:
- For any point \( P \) on the ellipse, the sum of the distances from \( P \) to the two foci \( F_1 \) and \( F_2 \) is equal to the length of the major axis, which is \( 2a \).
- Therefore, the length of the string is:
\[ \text{Length of the string} = 2a = 2 \times 3 = 6 \text{ cm} \]
- Additionally, we need to consider the distance from the point \( P \) to the foci:
\[ \text{Length of the string} = 2a + 2c = 6 + 2\sqrt{5} \text{ cm} \]

### Final Results:
- The necessary length of the string is \( 6 + 2\sqrt{5} \) cm.
- The distance between the pins is \( 2\sqrt{5} \) cm.