The radius of the circle whose centre is (-4,0) and which cuts the parabola `y^(2)=8x` at A and B such that the common chord AB subtends a right angle at the vertex of the parabola is equal to

Options

A. `4sqrt(13)`
B. `3sqrt(5)`
C. `3sqrt(2)`
D. `2sqrt(5)`

Explanation

To find the radius of the circle whose center is at (-4, 0) and which intersects the parabola \( y^2 = 8x \) at points A and B such that the chord AB subtends a right angle at the vertex of the parabola, we can follow these steps:

### Step 1: Understand the Parabola
The equation of the parabola is given as \( y^2 = 8x \). This is a standard form of a parabola that opens to the right. The vertex of this parabola is at the origin (0, 0).

### Step 2: Identify the Circle's Center
The center of the circle is given as (-4, 0).

### Step 3: Parametrize Points on the Parabola
We can represent points A and B on the parabola using a parameter \( t \):
- Point A: \( (at^2, 2at) \)
- Point B: \( (at^2, -2at) \)

Here, \( a \) is a constant related to the parabola. For the parabola \( y^2 = 8x \), we have \( 4a = 8 \) which gives \( a = 2 \). Thus, the coordinates of points A and B become:
- Point A: \( (2t^2, 4t) \)
- Point B: \( (2t^2, -4t) \)

### Step 4: Use the Right Angle Condition
The chord AB subtends a right angle at the vertex (0, 0). According to the property of the circle, if a chord subtends a right angle at the vertex of the parabola, then the following condition holds:
\[ VA^2 + VB^2 = AB^2 \]
where \( V \) is the vertex (0, 0).

### Step 5: Calculate Distances
1. Distance \( VA \):
\[ VA = \sqrt{(2t^2 - 0)^2 + (4t - 0)^2} = \sqrt{(2t^2)^2 + (4t)^2} = \sqrt{4t^4 + 16t^2} = 2t\sqrt{t^2 + 4} \]

2. Distance \( VB \):
\[ VB = \sqrt{(2t^2 - 0)^2 + (-4t - 0)^2} = \sqrt{(2t^2)^2 + (-4t)^2} = \sqrt{4t^4 + 16t^2} = 2t\sqrt{t^2 + 4} \]

3. Distance \( AB \):
\[ AB = \sqrt{(2t^2 - 2t^2)^2 + (4t - (-4t))^2} = \sqrt{(8t)^2} = 8t \]

### Step 6: Apply the Right Angle Condition
Using the right angle condition:
\[ VA^2 + VB^2 = AB^2 \]
\[ (2t\sqrt{t^2 + 4})^2 + (2t\sqrt{t^2 + 4})^2 = (8t)^2 \]
\[ 2(4t^2(t^2 + 4)) = 64t^2 \]
\[ 8t^2(t^2 + 4) = 64t^2 \]
Dividing both sides by \( 8t^2 \) (assuming \( t \neq 0 \)):
\[ t^2 + 4 = 8 \implies t^2 = 4 \implies t = 2 \text{ or } t = -2 \]

### Step 7: Find Coordinates of A and B
Using \( t = 2 \):
- Point A: \( (2(2^2), 4(2)) = (8, 8) \)
- Point B: \( (2(2^2), -4(2)) = (8, -8) \)

### Step 8: Calculate the Radius of the Circle
The radius \( r \) of the circle can be calculated using the distance from the center (-4, 0) to either point A or B:
\[ r = \sqrt{(8 - (-4))^2 + (8 - 0)^2} = \sqrt{(8 + 4)^2 + 8^2} = \sqrt{12^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} = 4\sqrt{13} \]

### Final Answer
The radius of the circle is \( 4\sqrt{13} \).

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