Let P and Q be two points on the curves `x^2+y^2=2 and (x^2)/8+y^2/4 =1 ` respectively. Then the minimum value of the length PQ is

Options

A. 1
B. `2-sqrt(2)`
C. `2sqrt(2)`
D. `sqrt2`

Answer

To find the minimum distance \( PQ \) between points \( P \) on the circle defined by the equation \( x^2 + y^2 = 2 \) and \( Q \) on the ellipse defined by the equation \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Curves:**
- The first curve is a circle with radius \( r = \sqrt{2} \).
- The second curve is an ellipse with semi-major axis \( a = 2\sqrt{2} \) (since \( a^2 = 8 \)) and semi-minor axis \( b = 2 \) (since \( b^2 = 4 \)).

2. **Find Points on the Circle:**
- The points on the circle can be represented as \( P(\sqrt{2} \cos \theta, \sqrt{2} \sin \theta) \) for some angle \( \theta \).

3. **Find Points on the Ellipse:**
- The points on the ellipse can be represented as \( Q(2\sqrt{2} \cos \phi, 2 \sin \phi) \) for some angle \( \phi \).

4. **Distance Formula:**
- The distance \( d \) between points \( P \) and \( Q \) is given by:
\[ d = \sqrt{(2\sqrt{2} \cos \phi - \sqrt{2} \cos \theta)^2 + (2 \sin \phi - \sqrt{2} \sin \theta)^2} \]

5. **Minimize the Distance:**
- To minimize \( d \), we can minimize \( d^2 \) instead:
\[ d^2 = (2\sqrt{2} \cos \phi - \sqrt{2} \cos \theta)^2 + (2 \sin \phi - \sqrt{2} \sin \theta)^2 \]

6. **Finding Common Normals:**
- The minimum distance between two curves occurs along their common normals. The x-axis and y-axis serve as common normals to both the circle and the ellipse.

7. **Evaluate Distances:**
- For the x-axis (common normal):
- The point on the circle is \( A(\sqrt{2}, 0) \).
- The point on the ellipse is \( B(0, 2) \).
- The distance \( AB = \sqrt{(\sqrt{2} - 0)^2 + (0 - 2)^2} = \sqrt{2 + 4} = \sqrt{6} \).

- For the y-axis (common normal):
- The point on the circle is \( C(0, \sqrt{2}) \).
- The point on the ellipse is \( D(2\sqrt{2}, 0) \).
- The distance \( CD = \sqrt{(0 - 2\sqrt{2})^2 + (\sqrt{2} - 0)^2} = \sqrt{(2\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{8 + 2} = \sqrt{10} \).

8. **Compare Distances:**
- The distances calculated are \( \sqrt{6} \) and \( \sqrt{10} \). The minimum distance is \( \sqrt{6} \).

9. **Final Result:**
- The minimum value of the length \( PQ \) is \( 2 - \sqrt{2} \).