An equilateral triangle is inscribed in the ellipse `x^2+3y^2=3` such that one vertex of the triangle is (0, 1) and one altitude of the triangle is along the y-axis. The length of its side is

Options

A. `4sqrt3//5`
B. `3sqrt3//5`
C. `6sqrt3//5`
D. `2sqrt3`

Answer

To find the length of the side of the equilateral triangle inscribed in the ellipse given by the equation \(x^2 + 3y^2 = 3\), we will follow these steps:

### Step 1: Rewrite the equation of the ellipse
The equation of the ellipse can be rewritten in standard form:
\[ \frac{x^2}{3} + \frac{y^2}{1} = 1 \]
This indicates that the semi-major axis \(a = \sqrt{3}\) and the semi-minor axis \(b = 1\).

### Step 2: Identify the vertex and altitude
We know one vertex of the triangle is at \(C(0, 1)\) and that one altitude is along the y-axis. This means the other two vertices \(A\) and \(B\) will be symmetric about the y-axis.

### Step 3: Determine the coordinates of points A and B
Let the coordinates of points \(A\) and \(B\) be \((-x, y_A)\) and \((x, y_A)\) respectively. Since the triangle is equilateral, the altitude from \(C\) to the base \(AB\) will bisect \(AB\) at point \(D(0, y_A)\).

### Step 4: Find the relationship between the coordinates
The length of the altitude \(CD\) can be calculated as:
\[ CD = 1 - y_A \]
The length of the side \(AB\) can be calculated using the distance formula:
\[ AB = \sqrt{((-x) - x)^2 + (y_A - y_A)^2} = 2x \]

### Step 5: Use the properties of the equilateral triangle
In an equilateral triangle, the relationship between the side length \(s\) and the altitude \(h\) is given by:
\[ h = \frac{\sqrt{3}}{2}s \]
Substituting \(s = 2x\) into the equation gives:
\[ 1 - y_A = \frac{\sqrt{3}}{2}(2x) \]
Thus, we have:
\[ 1 - y_A = \sqrt{3}x \]
From this, we can express \(y_A\) as:
\[ y_A = 1 - \sqrt{3}x \]

### Step 6: Substitute into the ellipse equation
Substituting \(y_A\) into the ellipse equation:
\[ x^2 + 3(1 - \sqrt{3}x)^2 = 3 \]
Expanding this:
\[ x^2 + 3(1 - 2\sqrt{3}x + 3x^2) = 3 \]
\[ x^2 + 3 - 6\sqrt{3}x + 9x^2 = 3 \]
Combining like terms:
\[ 10x^2 - 6\sqrt{3}x + 0 = 0 \]

### Step 7: Solve the quadratic equation
Factoring out \(2x\):
\[ 2x(5x - 3\sqrt{3}) = 0 \]
This gives us \(x = 0\) or \(x = \frac{3\sqrt{3}}{5}\).

### Step 8: Find the coordinates of A and B
Using \(x = \frac{3\sqrt{3}}{5}\):
\[ y_A = 1 - \sqrt{3}\left(\frac{3\sqrt{3}}{5}\right) = 1 - \frac{9}{5} = -\frac{4}{5} \]
Thus, the coordinates of \(A\) and \(B\) are:
\[ A\left(-\frac{3\sqrt{3}}{5}, -\frac{4}{5}\right), \quad B\left(\frac{3\sqrt{3}}{5}, -\frac{4}{5}\right) \]

### Step 9: Calculate the side length
The length of the side \(s\) is:
\[ s = 2x = 2\left(\frac{3\sqrt{3}}{5}\right) = \frac{6\sqrt{3}}{5} \]

### Final Answer
The length of the side of the equilateral triangle is:
\[ \frac{6\sqrt{3}}{5} \]