The points A(3, 6) and B lie on the parabola `y^(2)=4ax`, such that the chord AB subtends `90^(@)` at the origin, then the length of the chord AB is equal to

Options

A. `15sqrt(13)" units"`
B. `12sqrt(17)" units"`
C. `9sqrt(17)" units"`
D. `9sqrt(10)" units"`

Answer

To solve the problem step by step, we need to find the length of the chord AB that subtends a right angle at the origin, given that point A is (3, 6) and point B lies on the parabola defined by \(y^2 = 4ax\).

### Step 1: Identify the parabola and the coordinates of point B
The equation of the parabola is given as \(y^2 = 4ax\). Since point B lies on this parabola, we can express the coordinates of point B in terms of a parameter \(t\):
- The coordinates of point B can be represented as \(B(at^2, 2at)\).

### Step 2: Find the slopes of lines OA and OB
- The slope of line OA (from the origin O to point A) is calculated as:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 0}{3 - 0} = \frac{6}{3} = 2 \]

- The slope of line OB (from the origin O to point B) is:
\[ m_2 = \frac{2at - 0}{at^2 - 0} = \frac{2t}{t^2} = \frac{2}{t} \]

### Step 3: Use the condition for perpendicular lines
Since the chord AB subtends a right angle at the origin, the product of the slopes must equal -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes we found:
\[ 2 \cdot \frac{2}{t} = -1 \]
This simplifies to:
\[ \frac{4}{t} = -1 \implies t = -4 \]

### Step 4: Find the coordinates of point B
Now substituting \(t = -4\) into the equations for the coordinates of B:
\[ B(a(-4)^2, 2a(-4)) = B(16a, -8a) \]

### Step 5: Find the value of \(a\) using point A
Since point A (3, 6) lies on the parabola, we can substitute these values into the parabola equation:
\[ 6^2 = 4a \cdot 3 \implies 36 = 12a \implies a = 3 \]

### Step 6: Substitute \(a\) back to find coordinates of B
Now substituting \(a = 3\) into the coordinates of B:
\[ B(16 \cdot 3, -8 \cdot 3) = B(48, -24) \]

### Step 7: Calculate the length of chord AB
Using the distance formula to find the length of chord AB:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{(48 - 3)^2 + (-24 - 6)^2} = \sqrt{(45)^2 + (-30)^2} \]
Calculating:
\[ AB = \sqrt{2025 + 900} = \sqrt{2925} \]
Factoring out:
\[ 2925 = 225 \times 13 \implies AB = 15\sqrt{13} \]

### Final Answer
The length of the chord AB is \(15\sqrt{13}\) units.