Find the locus of the point of intersection of tangents to the circle `x=acostheta,y=asintheta` at the points whose parametric angles differ by `(i) pi/3, `

Options

A. Straight line
B. Ellipse
C. Circle is radius 2a
D. Circle of radiuis `(2a)/(sqrt(3))`

Answer

To find the locus of the point of intersection of tangents to the circle given by the parametric equations \( x = a \cos \theta \) and \( y = a \sin \theta \) at points whose parametric angles differ by \( \frac{\pi}{3} \), we will follow these steps:

### Step 1: Identify the Circle Equation
The given parametric equations represent a circle with the equation:
\[ x^2 + y^2 = a^2 \]

### Step 2: Determine the Points of Tangency
Let the first point of tangency be at angle \( \phi \):
\[ P_1 = (a \cos \phi, a \sin \phi) \]
The second point of tangency, which differs by \( \frac{\pi}{3} \), will be:
\[ P_2 = (a \cos(\phi + \frac{\pi}{3}), a \sin(\phi + \frac{\pi}{3})) \]

### Step 3: Write the Equations of the Tangents
The equation of the tangent to the circle at point \( P_1 \) is given by:
\[ x \cos \phi + y \sin \phi = a \]
The equation of the tangent at point \( P_2 \) is:
\[ x \cos(\phi + \frac{\pi}{3}) + y \sin(\phi + \frac{\pi}{3}) = a \]

### Step 4: Expand the Tangent Equation at \( P_2 \)
Using the angle addition formulas:
\[ \cos(\phi + \frac{\pi}{3}) = \cos \phi \cdot \frac{1}{2} - \sin \phi \cdot \frac{\sqrt{3}}{2} \]
\[ \sin(\phi + \frac{\pi}{3}) = \sin \phi \cdot \frac{1}{2} + \cos \phi \cdot \frac{\sqrt{3}}{2} \]
Thus, the tangent equation at \( P_2 \) becomes:
\[ x \left( \frac{1}{2} \cos \phi - \frac{\sqrt{3}}{2} \sin \phi \right) + y \left( \frac{1}{2} \sin \phi + \frac{\sqrt{3}}{2} \cos \phi \right) = a \]

### Step 5: Simplify the Tangent Equations
Let’s denote the coefficients:
\[ A = \frac{1}{2} \cos \phi - \frac{\sqrt{3}}{2} \sin \phi \]
\[ B = \frac{1}{2} \sin \phi + \frac{\sqrt{3}}{2} \cos \phi \]
Thus, the second tangent equation can be rewritten as:
\[ Ax + By = a \]

### Step 6: Find the Intersection of the Tangents
To find the intersection of the two tangents, we can set up the system of equations:
1. \( x \cos \phi + y \sin \phi = a \)
2. \( Ax + By = a \)

### Step 7: Solve the System of Equations
We can solve these two equations simultaneously to find the coordinates of the point of intersection \( (x, y) \).

### Step 8: Eliminate \( \phi \)
After solving, we will eliminate \( \phi \) to find the relationship between \( x \) and \( y \).

### Step 9: Derive the Locus Equation
After simplification, we will arrive at the locus equation:
\[ x^2 + y^2 = \frac{4a^2}{3} \]

### Conclusion
The locus of the point of intersection of the tangents to the circle at points whose parametric angles differ by \( \frac{\pi}{3} \) is given by:
\[ x^2 + y^2 = \frac{4a^2}{3} \]