The length of intercept cut but the line `4x+4sqrt3y-1=0` on the curve `y^(2)=4(x+(3)/(4))` is equal to

Options

A. 4 units
B. 9 units
C. 12 units
D. 16 units

Answer

To find the length of the intercept cut by the line \(4x + 4\sqrt{3}y - 1 = 0\) on the curve \(y^2 = 4\left(x + \frac{3}{4}\right)\), we can follow these steps:

### Step 1: Rewrite the equations
First, we rewrite the equations of the line and the parabola in a more manageable form.

The line can be rewritten as:
\[ y = -\frac{4}{4\sqrt{3}}x + \frac{1}{4\sqrt{3}} = -\frac{1}{\sqrt{3}}x + \frac{1}{4\sqrt{3}} \]

The parabola is already in the standard form:
\[ y^2 = 4\left(x + \frac{3}{4}\right) \]

### Step 2: Substitute the line equation into the parabola equation
Next, we substitute the expression for \(y\) from the line equation into the parabola equation.

Substituting \(y = -\frac{1}{\sqrt{3}}x + \frac{1}{4\sqrt{3}}\) into \(y^2 = 4\left(x + \frac{3}{4}\right)\):
\[ \left(-\frac{1}{\sqrt{3}}x + \frac{1}{4\sqrt{3}}\right)^2 = 4\left(x + \frac{3}{4}\right) \]

### Step 3: Expand and simplify
Expanding the left side:
\[ \frac{1}{3}x^2 - \frac{1}{6}x + \frac{1}{48} = 4x + 3 \]

Now, multiply through by 48 to eliminate the fractions:
\[ 16x^2 - 8x + 1 = 192x + 144 \]

Rearranging gives:
\[ 16x^2 - 200x - 143 = 0 \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 16\), \(b = -200\), and \(c = -143\).

Calculating the discriminant:
\[ b^2 - 4ac = (-200)^2 - 4 \cdot 16 \cdot (-143) = 40000 + 9152 = 49152 \]

Now, applying the quadratic formula:
\[ x = \frac{200 \pm \sqrt{49152}}{32} \]

### Step 5: Find the length of the intercept
The two \(x\) values found will give us the points where the line intersects the parabola. The length of the intercept can be calculated as the difference between these two \(x\) values.

Let \(x_1\) and \(x_2\) be the roots of the quadratic equation. The length of the intercept \(L\) is given by:
\[ L = |x_1 - x_2| = \frac{\sqrt{49152}}{16} \]

Calculating \(\sqrt{49152}\):
\[ \sqrt{49152} = 16\sqrt{192} = 16 \cdot 8\sqrt{3} = 128\sqrt{3} \]

Thus, the length of the intercept is:
\[ L = \frac{128\sqrt{3}}{16} = 8\sqrt{3} \]

### Final Answer
The length of the intercept cut by the line on the curve is \(8\sqrt{3}\).

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