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 value of the length \( PQ \) where \( P \) is a point on the circle defined by \( x^2 + y^2 = 2 \) and \( Q \) is a point on the ellipse defined by \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \), we can follow these steps:

### Step 1: Identify the points on the curves
The circle \( x^2 + y^2 = 2 \) has a radius of \( \sqrt{2} \) and is centered at the origin. The ellipse \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \) has semi-major axis \( 2\sqrt{2} \) along the x-axis and semi-minor axis \( 2 \) along the y-axis.

### Step 2: Parametrize the points
Let \( P \) be a point on the circle. We can express \( P \) in parametric form as:
\[ P = (\sqrt{2} \cos \theta, \sqrt{2} \sin \theta) \]
for \( \theta \in [0, 2\pi) \).

Let \( Q \) be a point on the ellipse. We can express \( Q \) in parametric form as:
\[ Q = (2\sqrt{2} \cos \phi, 2 \sin \phi) \]
for \( \phi \in [0, 2\pi) \).

### Step 3: Calculate the distance \( PQ \)
The distance \( d \) between points \( P \) and \( Q \) is given by:
\[ d = \sqrt{(2\sqrt{2} \cos \phi - \sqrt{2} \cos \theta)^2 + (2 \sin \phi - \sqrt{2} \sin \theta)^2} \]

### Step 4: Simplify the distance formula
Expanding the distance formula:
\[ d^2 = (2\sqrt{2} \cos \phi - \sqrt{2} \cos \theta)^2 + (2 \sin \phi - \sqrt{2} \sin \theta)^2 \]
This can be simplified further, but we will focus on minimizing \( d \).

### Step 5: Use calculus or geometric interpretation
To find the minimum distance, we can analyze the geometry of the situation. The shortest distance between a point on the circle and a point on the ellipse occurs along the line that is normal to both curves at points \( P \) and \( Q \).

### Step 6: Find the minimum distance
By symmetry and the properties of the curves, the minimum distance occurs when the line connecting \( P \) and \( Q \) is perpendicular to the tangent lines at those points.

After analyzing the geometry or using calculus, we find that the minimum distance \( PQ \) is:
\[ \text{Minimum distance} = 2 \]

### Conclusion
Thus, the minimum value of the length \( PQ \) is \( 2 \).