The ellipse `4x^(2)+8y^(2)=64 and ` the circle `x^(2)+y^(2)=9` intersect at points where the y-coordinates is

Options

A. `+-sqrt(2)`
B. `+-sqrt(5)`
C. `+-sqrt(6)`
D. `+-sqrt(7)`

Explanation

To find the y-coordinates where the ellipse \(4x^2 + 8y^2 = 64\) and the circle \(x^2 + y^2 = 9\) intersect, we can follow these steps:

### Step 1: Write down the equations
The equations of the ellipse and the circle are:
- Ellipse: \(4x^2 + 8y^2 = 64\)
- Circle: \(x^2 + y^2 = 9\)

### Step 2: Simplify the equations
First, we can simplify the equation of the ellipse by dividing everything by 64:
\[ \frac{4x^2}{64} + \frac{8y^2}{64} = 1 \implies \frac{x^2}{16} + \frac{y^2}{8} = 1 \]
This shows the ellipse is centered at the origin with semi-major axis \(4\) along the x-axis and semi-minor axis \(2\sqrt{2}\) along the y-axis.

### Step 3: Express \(x^2\) in terms of \(y^2\) using the circle's equation
From the circle's equation \(x^2 + y^2 = 9\), we can express \(x^2\) as:
\[ x^2 = 9 - y^2 \]

### Step 4: Substitute \(x^2\) into the ellipse's equation
Now, substitute \(x^2\) into the ellipse's equation:
\[ 4(9 - y^2) + 8y^2 = 64 \]

### Step 5: Simplify the equation
Expanding and simplifying gives:
\[ 36 - 4y^2 + 8y^2 = 64 \]
Combine like terms:
\[ 36 + 4y^2 = 64 \]

### Step 6: Solve for \(y^2\)
Rearranging the equation:
\[ 4y^2 = 64 - 36 \]
\[ 4y^2 = 28 \]
Dividing by 4:
\[ y^2 = 7 \]

### Step 7: Find \(y\)
Taking the square root of both sides gives:
\[ y = \pm \sqrt{7} \]

### Final Answer
The y-coordinates at which the ellipse and the circle intersect are:
\[ y = \sqrt{7} \quad \text{and} \quad y = -\sqrt{7} \]