If the line `y=x sqrt3 - 3` cuts the parabola `y^2 = x+2` at Pand Q and if A be the point `(sqrt3,0)`, then `AP. AQ` is

Options

A. `2/3(sqrt3+2)`
B. `4/3(sqrt3+2)`
C. `4/3(2-sqrt3)`
D. `2sqrt3`

Answer

To solve the problem, we need to find the product of the distances from point A to points P and Q where the line intersects the parabola. Let's break down the solution step by step.

### Step 1: Identify the equations
We have the line given by:
\[ y = x \sqrt{3} - 3 \]
And the parabola given by:
\[ y^2 = x + 2 \]

### Step 2: Substitute the line equation into the parabola equation
To find the points of intersection (P and Q), we substitute the line equation into the parabola equation:
\[ (x \sqrt{3} - 3)^2 = x + 2 \]

### Step 3: Expand and simplify the equation
Expanding the left side:
\[ (x \sqrt{3} - 3)^2 = 3x^2 - 6\sqrt{3}x + 9 \]
Setting this equal to the right side:
\[ 3x^2 - 6\sqrt{3}x + 9 = x + 2 \]

Now, rearranging the equation:
\[ 3x^2 - 6\sqrt{3}x + 9 - x - 2 = 0 \]
This simplifies to:
\[ 3x^2 - (6\sqrt{3} + 1)x + 7 = 0 \]

### Step 4: Use the quadratic formula to find x-coordinates of P and Q
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3 \), \( b = -(6\sqrt{3} + 1) \), and \( c = 7 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-(6\sqrt{3} + 1))^2 - 4 \cdot 3 \cdot 7 \]
\[ = (6\sqrt{3} + 1)^2 - 84 \]
\[ = 108 + 12\sqrt{3} + 1 - 84 \]
\[ = 25 + 12\sqrt{3} \]

Now substituting back into the quadratic formula:
\[ x = \frac{6\sqrt{3} + 1 \pm \sqrt{25 + 12\sqrt{3}}}{6} \]

### Step 5: Find the y-coordinates of points P and Q
Substituting the x-coordinates back into the line equation to find the corresponding y-coordinates.

### Step 6: Calculate distances AP and AQ
Let \( A = (\sqrt{3}, 0) \). The distances \( AP \) and \( AQ \) can be calculated using the distance formula:
\[ AP = \sqrt{(x_P - \sqrt{3})^2 + (y_P - 0)^2} \]
\[ AQ = \sqrt{(x_Q - \sqrt{3})^2 + (y_Q - 0)^2} \]

### Step 7: Find the product \( AP \cdot AQ \)
Finally, we compute:
\[ AP \cdot AQ \]