In how many points do the ellipse `x^(2)/(4) + y^(2)/(8) = 1` and the circle `x^2 +y^2 = 9` intersect?

Options

A. One
B. Two
C. Four
D. None of the above

Answer

To find the number of intersection points between the ellipse given by the equation \( \frac{x^2}{4} + \frac{y^2}{8} = 1 \) and the circle given by the equation \( x^2 + y^2 = 9 \), we can follow these steps:

### Step 1: Rewrite the equations
We have the equations:
1. Ellipse: \( \frac{x^2}{4} + \frac{y^2}{8} = 1 \)
2. Circle: \( x^2 + y^2 = 9 \)

### Step 2: Express \( y^2 \) in terms of \( x^2 \) from the circle
From the circle's equation, we can express \( y^2 \):
\[ y^2 = 9 - x^2 \]

### Step 3: Substitute \( y^2 \) into the ellipse equation
Now, substitute \( y^2 \) from the circle's equation into the ellipse's equation:
\[ \frac{x^2}{4} + \frac{9 - x^2}{8} = 1 \]

### Step 4: Find a common denominator and simplify
The common denominator for the fractions is 8. Rewrite the equation:
\[ \frac{2x^2}{8} + \frac{9 - x^2}{8} = 1 \]
Combine the terms:
\[ \frac{2x^2 + 9 - x^2}{8} = 1 \]
This simplifies to:
\[ \frac{x^2 + 9}{8} = 1 \]

### Step 5: Clear the fraction by multiplying by 8
Multiply both sides by 8:
\[ x^2 + 9 = 8 \]

### Step 6: Solve for \( x^2 \)
Rearranging gives:
\[ x^2 = 8 - 9 \]
\[ x^2 = -1 \]

### Step 7: Analyze the result
Since \( x^2 = -1 \) is not possible (as the square of a real number cannot be negative), this means there are no real solutions for \( x \).

### Conclusion
Thus, the ellipse and the circle do not intersect at any points. Therefore, the number of intersection points is **0**.