The locus of the point of intersection of the tangents drawn to the ellipse `(x^(2))/(4)+y^(2)=1` if the difference of the eccentric angle of their contanct is `(2pi)/(3)` is

Options

A. `(x^(2))/(4)+y^(2)=4`
B. `(x^(2))/(4)+y^(2)=1`
C. `4x^(2)+y^(2)=16`
D. none of these

Answer

To find the locus of the point of intersection of the tangents drawn to the ellipse \(\frac{x^2}{4} + y^2 = 1\) when the difference of the eccentric angles of their points of contact is \(\frac{2\pi}{3}\), we can follow these steps:

### Step 1: Identify the ellipse parameters
The given ellipse is in the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 4\) and \(b^2 = 1\). Thus, we have:
- \(a = 2\)
- \(b = 1\)

### Step 2: Define the points of contact
Let the points of contact on the ellipse be \(P\) and \(Q\) corresponding to the eccentric angles \(\theta_1\) and \(\theta_2\). The coordinates of these points are:
- \(P(a \cos \theta_1, b \sin \theta_1) = (2 \cos \theta_1, \sin \theta_1)\)
- \(Q(a \cos \theta_2, b \sin \theta_2) = (2 \cos \theta_2, \sin \theta_2)\)

### Step 3: Write the equations of the tangents
The equation of the tangent at point \(P\) is given by:
\[ \frac{x \cos \theta_1}{2} + \frac{y \sin \theta_1}{1} = 1 \]
This simplifies to:
\[ x \cos \theta_1 + 2y \sin \theta_1 = 2 \quad \text{(1)} \]

The equation of the tangent at point \(Q\) is:
\[ \frac{x \cos \theta_2}{2} + \frac{y \sin \theta_2}{1} = 1 \]
This simplifies to:
\[ x \cos \theta_2 + 2y \sin \theta_2 = 2 \quad \text{(2)} \]

### Step 4: Find the point of intersection
To find the intersection of the two tangents, we solve equations (1) and (2) simultaneously.

From (1):
\[ x \cos \theta_1 + 2y \sin \theta_1 = 2 \]
From (2):
\[ x \cos \theta_2 + 2y \sin \theta_2 = 2 \]

### Step 5: Subtract the equations
Subtract equation (1) from equation (2):
\[ (x \cos \theta_2 - x \cos \theta_1) + 2y (\sin \theta_2 - \sin \theta_1) = 0 \]
This gives:
\[ x (\cos \theta_2 - \cos \theta_1) = -2y (\sin \theta_2 - \sin \theta_1) \]

### Step 6: Use the given condition
We know that \(|\theta_2 - \theta_1| = \frac{2\pi}{3}\). Using the sine and cosine difference formulas, we can express the above equation in terms of the known angles.

### Step 7: Substitute values
Using the sine and cosine values for \(\frac{2\pi}{3}\):
- \(\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\)
- \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\)

### Step 8: Find the locus
After substituting and simplifying, we will arrive at the locus equation. The final locus will be of the form:
\[ \frac{x^2}{4} + y^2 = 4 \]

### Conclusion
Thus, the locus of the point of intersection of the tangents is:
\[ \frac{x^2}{4} + y^2 = 4 \]