The length of the chord of the parabola `y^(2) = 12x` passing through the vertex and making an angle of `60^(@)` with the axis of x is

Options

A. `(8)/(3)`
B. 8
C. `(16)/(3)`
D. 4

Answer

To find the length of the chord of the parabola \( y^2 = 12x \) that passes through the vertex and makes an angle of \( 60^\circ \) with the x-axis, we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 12x \). We can compare this with the standard form \( y^2 = 4ax \). Here, we have:
\[ 4a = 12 \implies a = 3 \]

### Step 2: Parametric equations of the parabola
The parametric equations for the parabola \( y^2 = 12x \) can be expressed as:
\[ x = at^2 = 3t^2, \quad y = 2at = 6t \]

### Step 3: Determine the slope of the chord
The chord makes an angle of \( 60^\circ \) with the x-axis. The slope \( m \) of the chord can be calculated as:
\[ m = \tan(60^\circ) = \sqrt{3} \]

### Step 4: Equation of the chord
Since the chord passes through the vertex (0, 0) and has a slope of \( \sqrt{3} \), the equation of the chord can be written as:
\[ y = \sqrt{3}x \]

### Step 5: Find the points of intersection
To find the points of intersection of the chord with the parabola, substitute \( y = \sqrt{3}x \) into the parabola's equation:
\[ (\sqrt{3}x)^2 = 12x \]
\[ 3x^2 = 12x \]
\[ 3x^2 - 12x = 0 \]
Factoring out \( 3x \):
\[ 3x(x - 4) = 0 \]
Thus, \( x = 0 \) or \( x = 4 \).

### Step 6: Find corresponding y-coordinates
For \( x = 0 \):
\[ y = \sqrt{3}(0) = 0 \]
For \( x = 4 \):
\[ y = \sqrt{3}(4) = 4\sqrt{3} \]

### Step 7: Identify the points of intersection
The points of intersection are:
1. \( A(0, 0) \)
2. \( B(4, 4\sqrt{3}) \)

### Step 8: Calculate the length of the chord
The length of the chord \( AB \) can be calculated using the distance formula:
\[ L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ L = \sqrt{(4 - 0)^2 + (4\sqrt{3} - 0)^2} \]
\[ = \sqrt{4^2 + (4\sqrt{3})^2} \]
\[ = \sqrt{16 + 48} = \sqrt{64} = 8 \]

### Final Answer
The length of the chord is \( 8 \) units.

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