The common tangent of `x^(2)+y^(2)=4 and 2x^(2)+y^(2)=2` is

Options

A. x+y+4=0
B. x-y+7=0
C. 2x+3y+8=0
D. None of these

Answer

To find the common tangent of the given circle and ellipse, we will follow these steps:

### Step 1: Identify the equations of the circle and ellipse.
The equations given are:
1. Circle: \( x^2 + y^2 = 4 \)
2. Ellipse: \( 2x^2 + y^2 = 2 \)

### Step 2: Rewrite the ellipse equation in standard form.
To rewrite the ellipse equation, we divide the entire equation by 2:
\[ \frac{x^2}{1} + \frac{y^2}{2} = 1 \]
This shows that \( a^2 = 1 \) and \( b^2 = 2 \), where \( a = 1 \) and \( b = \sqrt{2} \).

### Step 3: Determine the radius of the circle.
From the circle equation \( x^2 + y^2 = 4 \), we can see that the radius \( r \) is:
\[ r = \sqrt{4} = 2 \]

### Step 4: Write the tangent equations for both the circle and the ellipse.
The general form of the tangent line to a circle in slope-intercept form is:
\[ y = mx \pm r\sqrt{1 + m^2} \]
For our circle, this becomes:
\[ y = mx \pm 2\sqrt{1 + m^2} \]

For the ellipse, the tangent line in slope-intercept form is:
\[ y = mx \pm \sqrt{a^2m^2 + b^2} \]
Substituting \( a^2 = 1 \) and \( b^2 = 2 \), we have:
\[ y = mx \pm \sqrt{m^2 + 2} \]

### Step 5: Set the two tangent equations equal to each other.
To find the common tangents, we equate the two expressions:
\[ 2\sqrt{1 + m^2} = \sqrt{m^2 + 2} \]

### Step 6: Square both sides to eliminate the square roots.
Squaring both sides gives:
\[ 4(1 + m^2) = m^2 + 2 \]
Expanding this results in:
\[ 4 + 4m^2 = m^2 + 2 \]

### Step 7: Rearrange the equation.
Rearranging gives:
\[ 4m^2 - m^2 + 4 - 2 = 0 \]
This simplifies to:
\[ 3m^2 + 2 = 0 \]

### Step 8: Solve for \( m^2 \).
From \( 3m^2 + 2 = 0 \), we find:
\[ 3m^2 = -2 \quad \Rightarrow \quad m^2 = -\frac{2}{3} \]
Since \( m^2 \) cannot be negative, this indicates that there are no real solutions for \( m \).

### Conclusion:
Since there are no valid values for \( m \), we conclude that there are no common tangents between the circle and the ellipse.

### Final Answer:
**There are no common tangents.**

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