A circle with radius `|a|` and center on the y-axis slied along it and a variable line through (a, 0) cuts the circle at points `Pa n dQ` . The region in which the point of intersection of the tangents to the circle at points `P` and `Q` lies is represented by `y^2geq4(a x-a^2)` (b) `y^2lt=4(a x-a^2)` `ygeq4(a x-a^2)` (d) `ylt=4(a x-a^2)`

Options

A. `y^(2) ge 4a (x-a)`
B. `y^(2) le 4ax`
C. `x^(2) +y^(2) le 4a^(2)`
D. `x^(2) -y^(2) ge a^(2)`

Explanation

To solve the problem step by step, we will analyze the given information about the circle and the line, and derive the required conditions for the tangents.

### Step 1: Define the Circle
The circle has a center on the y-axis and a radius of |a|. Therefore, we can represent the center of the circle as \( (0, \alpha) \), where \( \alpha \) is some real number. The equation of the circle can be written as:
\[ x^2 + (y - \alpha)^2 = a^2 \]
Expanding this, we get:
\[ x^2 + y^2 - 2y\alpha + \alpha^2 = a^2 \]
Rearranging gives us:
\[ x^2 + y^2 - 2y\alpha + (\alpha^2 - a^2) = 0 \tag{1} \]

### Step 2: Define the Line
We are given a variable line that passes through the point \( (a, 0) \) and intersects the circle at points \( P \) and \( Q \). Let’s denote the coordinates of point \( P \) as \( (h, k) \).

### Step 3: Equation of the Chord of Contact
The chord of contact for the circle at point \( (h, k) \) can be expressed using the formula:
\[ xx_1 + yy_1 - a^2 = 0 \]
Substituting \( (h, k) \) into the equation gives:
\[ xh + yk - a^2 = 0 \tag{2} \]

### Step 4: Substitute the Point of Intersection
Since the line passes through the point \( (a, 0) \), we substitute this point into equation (2):
\[ a \cdot h + 0 \cdot k - a^2 = 0 \]
This simplifies to:
\[ ah - a^2 = 0 \implies h = a \tag{3} \]

### Step 5: Condition for Tangents
The tangents to the circle at points \( P \) and \( Q \) will intersect at a point. For the point of intersection of the tangents to be real, the discriminant of the quadratic equation formed must be non-negative. The quadratic equation derived from the circle's equation and the line's equation is:
\[ k^2 - 4(h - a^2) \geq 0 \]
Substituting \( h = a \) from equation (3) gives:
\[ k^2 - 4(a - a^2) \geq 0 \]
This simplifies to:
\[ k^2 \geq 4(a - a^2) \tag{4} \]

### Step 6: Rearranging the Condition
Rearranging equation (4) gives us:
\[ k^2 \geq 4a(1 - a) \]
This can be rewritten as:
\[ k^2 \geq 4a(x - a^2) \]
where \( x \) is the x-coordinate corresponding to the line.

### Conclusion
Thus, the region in which the point of intersection of the tangents to the circle at points \( P \) and \( Q \) lies is represented by:
\[ y^2 \geq 4(a x - a^2) \]
The correct option is (a).