The length of the chord `y=sqrt3x-2sqrt3` intercepted by the parabola `y^(2)=4(x-1)` is equal to

Options

A. `4sqrt3` units
B. `(8)/(3)` units
C. `(16)/(3)` units
D. `(4)/(sqrt3)` units

Answer

To find the length of the chord intercepted by the parabola \( y^2 = 4(x - 1) \) and the line \( y = \sqrt{3}x - 2\sqrt{3} \), we will follow these steps:

### Step 1: Substitute the line equation into the parabola equation

We start by substituting \( y = \sqrt{3}x - 2\sqrt{3} \) into the parabola equation \( y^2 = 4(x - 1) \).

\[ (\sqrt{3}x - 2\sqrt{3})^2 = 4(x - 1) \]

### Step 2: Expand the left-hand side

Expanding the left-hand side gives us:

\[ 3x^2 - 4\sqrt{3} \cdot 2\sqrt{3} x + 4 \cdot 3 = 4(x - 1) \]

This simplifies to:

\[ 3x^2 - 8\sqrt{3}x + 12 = 4x - 4 \]

### Step 3: Rearrange the equation

Now, we rearrange the equation to set it to zero:

\[ 3x^2 - 8\sqrt{3}x - 4x + 12 + 4 = 0 \]

This simplifies to:

\[ 3x^2 - (8\sqrt{3} + 4)x + 16 = 0 \]

### Step 4: Use the quadratic formula to find the roots

We can now use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = -(8\sqrt{3} + 4) \), and \( c = 16 \).

Calculating the discriminant:

\[ b^2 - 4ac = (8\sqrt{3} + 4)^2 - 4 \cdot 3 \cdot 16 \]

Calculating \( (8\sqrt{3} + 4)^2 \):

\[ = 192 + 64\sqrt{3} + 16 = 208 + 64\sqrt{3} \]

Calculating \( 4 \cdot 3 \cdot 16 = 192 \).

Thus, the discriminant becomes:

\[ (208 + 64\sqrt{3}) - 192 = 16 + 64\sqrt{3} \]

### Step 5: Calculate the roots

Now substituting back into the quadratic formula:

\[ x = \frac{8\sqrt{3} + 4 \pm \sqrt{16 + 64\sqrt{3}}}{6} \]

### Step 6: Find the corresponding y values

For each \( x \) value found, substitute back into the line equation \( y = \sqrt{3}x - 2\sqrt{3} \) to find the corresponding \( y \) values.

### Step 7: Calculate the distance between the two points

Let the two points be \( A(x_1, y_1) \) and \( B(x_2, y_2) \). The length of the chord \( AB \) can be calculated using the distance formula:

\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Final Calculation

After calculating the \( x \) and \( y \) values, substitute them into the distance formula to find the length of the chord.