Through the vertex `O` of the parabola `y^2=4a x` , two chords `O Pa n dO Q` are drawn and the circles on OP and OQ as diameters intersect at `Rdot` If `theta_1,theta_2` , and `varphi` are the angles made with the axis by the tangents at `P` and `Q` on the parabola and by `O R ,` then value of `cottheta_1+cottheta_2` is (a) `-2tanvarphi` (b) `-2tan(pi-varphi)` (c) 0 (d) `2cotvarphi`

Options

A. `-2tanphi`
B. `-2tan(pi-phi)`
C. 0
D. `2cotphi`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the parabola and its properties
The given parabola is \( y^2 = 4ax \). The vertex \( O \) of the parabola is at the origin \( (0, 0) \).

### Step 2: Define points on the parabola
Let \( P \) and \( Q \) be points on the parabola with parameters \( t_1 \) and \( t_2 \) respectively. The coordinates of these points can be expressed as:
- \( P(t_1) = (at_1^2, 2at_1) \)
- \( Q(t_2) = (at_2^2, 2at_2) \)

### Step 3: Write the equations of the tangents at points \( P \) and \( Q \)
The equation of the tangent at point \( P \) is given by:
\[ t_1y = x + at_1^2 \quad \Rightarrow \quad y = \frac{1}{t_1}x + at_1 \]
The slope \( m_1 \) of this tangent is \( \frac{1}{t_1} \).

Similarly, for point \( Q \):
\[ t_2y = x + at_2^2 \quad \Rightarrow \quad y = \frac{1}{t_2}x + at_2 \]
The slope \( m_2 \) of this tangent is \( \frac{1}{t_2} \).

### Step 4: Relate slopes to angles
The angles \( \theta_1 \) and \( \theta_2 \) made by the tangents with the x-axis are related to their slopes:
\[ \tan \theta_1 = \frac{1}{t_1} \quad \text{and} \quad \tan \theta_2 = \frac{1}{t_2} \]
Thus, we can express cotangents:
\[ \cot \theta_1 = t_1 \quad \text{and} \quad \cot \theta_2 = t_2 \]

### Step 5: Find \( \cot \theta_1 + \cot \theta_2 \)
From the above, we have:
\[ \cot \theta_1 + \cot \theta_2 = t_1 + t_2 \]

### Step 6: Write the equations of the circles
The equation of the circle with diameter \( OP \) is:
\[ x^2 + y^2 - at_1^2x - 2at_1y = 0 \]
The equation of the circle with diameter \( OQ \) is:
\[ x^2 + y^2 - at_2^2x - 2at_2y = 0 \]

### Step 7: Find the equation of the common chord
To find the equation of the common chord, we subtract the two circle equations:
\[ (at_2^2 - at_1^2)x + 2a(t_2 - t_1)y = 0 \]
This simplifies to:
\[ (t_2^2 - t_1^2)x + 2(t_2 - t_1)y = 0 \]
From this, we can express \( y \) in terms of \( x \):
\[ y = -\frac{t_2^2 - t_1^2}{2(t_2 - t_1)}x \]

### Step 8: Relate the angles
The angle \( \varphi \) made by the line \( OR \) with the x-axis is given by:
\[ \tan \varphi = -\frac{t_2^2 - t_1^2}{2(t_2 + t_1)} \]
Thus, we can relate \( \cot \theta_1 + \cot \theta_2 \) to \( \tan \varphi \):
\[ \cot \theta_1 + \cot \theta_2 = t_1 + t_2 = -2 \tan \varphi \]

### Conclusion
Therefore, the value of \( \cot \theta_1 + \cot \theta_2 \) is:
\[ \cot \theta_1 + \cot \theta_2 = -2 \tan \varphi \]
The correct option is (a) \( -2 \tan \varphi \).