If the major axis of an ellipse is alongthe y-axis and it passes through the points `(0, sqrt(3))` and `(sqrt(2), 0)`, then the equation of the elliipse is

Options

A. `3x^(2) + 2y^(2) =1`
B. `3x^(2) + 2y^(2) = 6`
C. `2x^(2) + 3y^(2) = 1`
D. `2x^(2) + 3y^(2) = 6`

Answer

To find the equation of the ellipse given that its major axis is along the y-axis and it passes through the points \( (0, \sqrt{3}) \) and \( (\sqrt{2}, 0) \), we can follow these steps:

### Step 1: Write the standard form of the ellipse
Since the major axis is along the y-axis, the standard form of the ellipse is given by:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \( b^2 > a^2 \).

### Step 2: Substitute the first point \( (0, \sqrt{3}) \)
Substituting the point \( (0, \sqrt{3}) \) into the equation:

\[ \frac{0^2}{a^2} + \frac{(\sqrt{3})^2}{b^2} = 1 \]

This simplifies to:

\[ \frac{3}{b^2} = 1 \]

From this, we can solve for \( b^2 \):

\[ b^2 = 3 \]

### Step 3: Substitute the second point \( (\sqrt{2}, 0) \)
Now, substituting the point \( (\sqrt{2}, 0) \) into the equation:

\[ \frac{(\sqrt{2})^2}{a^2} + \frac{0^2}{b^2} = 1 \]

This simplifies to:

\[ \frac{2}{a^2} = 1 \]

From this, we can solve for \( a^2 \):

\[ a^2 = 2 \]

### Step 4: Write the equation of the ellipse
Now that we have \( a^2 \) and \( b^2 \), we can write the equation of the ellipse:

\[ \frac{x^2}{2} + \frac{y^2}{3} = 1 \]

### Step 5: Rearranging to standard form
To express this in a more standard form, we can multiply through by the least common multiple of the denominators (which is 6):

\[ 6 \left( \frac{x^2}{2} + \frac{y^2}{3} \right) = 6 \]

This gives:

\[ 3x^2 + 2y^2 = 6 \]

### Final Equation
Thus, the equation of the ellipse is:

\[ 3x^2 + 2y^2 = 6 \]

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