The distance of the point `(sqrt6 cos theta , sqrt2 sin theta)` on the ellipse from the centre of the ellipse is 2 if `theta` =

Options

A. `pi//6`
B. `pi//4`
C. `pi//3`
D. none of these

Answer

To solve the problem step by step, we will find the value of \( \theta \) such that the distance of the point \( (\sqrt{6} \cos \theta, \sqrt{2} \sin \theta) \) from the center of the ellipse (which is at the origin \( (0, 0) \)) is equal to 2.

### Step 1: Understand the Distance Formula
The distance \( d \) between a point \( (x_1, y_1) \) and another point \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, the point is \( (\sqrt{6} \cos \theta, \sqrt{2} \sin \theta) \) and the center of the ellipse is \( (0, 0) \).

### Step 2: Set Up the Distance Equation
We want the distance from the point to the center to be equal to 2:
\[ \sqrt{(\sqrt{6} \cos \theta - 0)^2 + (\sqrt{2} \sin \theta - 0)^2} = 2 \]
This simplifies to:
\[ \sqrt{6 \cos^2 \theta + 2 \sin^2 \theta} = 2 \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ 6 \cos^2 \theta + 2 \sin^2 \theta = 4 \]

### Step 4: Substitute \( \sin^2 \theta \)
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, we can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Substituting this into the equation gives:
\[ 6 \cos^2 \theta + 2(1 - \cos^2 \theta) = 4 \]

### Step 5: Simplify the Equation
Now, simplify the equation:
\[ 6 \cos^2 \theta + 2 - 2 \cos^2 \theta = 4 \]
This simplifies to:
\[ 4 \cos^2 \theta + 2 = 4 \]
Subtracting 2 from both sides:
\[ 4 \cos^2 \theta = 2 \]
Dividing by 4:
\[ \cos^2 \theta = \frac{1}{2} \]

### Step 6: Solve for \( \theta \)
Taking the square root of both sides:
\[ \cos \theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]
This gives us two possible angles:
\[ \theta = \frac{\pi}{4} \quad \text{or} \quad \theta = \frac{7\pi}{4} \]

### Final Answer
Thus, the values of \( \theta \) that satisfy the condition are:
\[ \theta = \frac{\pi}{4} \quad \text{or} \quad \theta = \frac{7\pi}{4} \]