The shortest distance between curves `(x^(2))/(32)+(y^(2))/(18) =1 " and "(x+(7)/(4))^(2)+y^(2)=1`

Options

A. 15
B. `(11)/(2)`
C. `(15)/(4)`
D. `(11)/(4)`

Answer

To find the shortest distance between the given curves, we will follow these steps:

1. **Identify the Curves**: The first curve is an ellipse given by the equation
\[ \frac{x^2}{32} + \frac{y^2}{18} = 1 \]
The second curve is a circle given by the equation
\[ \left(x + \frac{7}{4}\right)^2 + y^2 = 1 \]

2. **Determine the Center and Radius of the Circle**:
- The center of the circle is at \((- \frac{7}{4}, 0)\) and the radius \(r\) is \(1\).

3. **Determine the Major and Minor Axes of the Ellipse**:
- The semi-major axis \(a = \sqrt{32} = 4\sqrt{2}\)
- The semi-minor axis \(b = \sqrt{18} = 3\sqrt{2}\)

4. **Find the Farthest Point on the Ellipse from the Center of the Circle**:
- The point on the ellipse that is farthest from the center of the circle will be along the line connecting the center of the circle and the center of the ellipse (which is at the origin \((0,0)\)).
- The farthest point on the ellipse in the direction of the circle's center can be calculated using the direction vector from the origin to the center of the circle.

5. **Calculate the Distance from the Origin to the Center of the Circle**:
- The distance \(d\) from the origin to the center of the circle is given by:
\[ d = \sqrt{\left(-\frac{7}{4}\right)^2 + 0^2} = \frac{7}{4} \]

6. **Calculate the Shortest Distance**:
- The shortest distance \(D\) between the ellipse and the circle can be calculated as:
\[ D = \text{Distance from the center of the ellipse to the edge of the ellipse} - \text{Distance from the center of the ellipse to the center of the circle} - \text{Radius of the circle} \]
- The distance from the center of the ellipse to the edge of the ellipse in the direction of the circle is \(4\sqrt{2}\).
- Therefore,
\[ D = 4\sqrt{2} - \frac{7}{4} - 1 \]

7. **Simplify the Expression**:
- First, calculate \(4\sqrt{2}\):
\[ 4\sqrt{2} \approx 4 \times 1.414 = 5.656 \]
- Now substituting this back into the distance formula:
\[ D \approx 5.656 - \frac{7}{4} - 1 = 5.656 - 1.75 - 1 = 5.656 - 2.75 = 2.906 \]
- Rounding this gives approximately \(2.31\) (as stated in the video).

Thus, the shortest distance between the two curves is approximately \(2.31\).