Question

The line \( y - \sqrt{3}x + 3 = 0 \) cuts the parabola \( y^2 = x + 2 \) at the points \( P \) and \( Q \). If the co-ordinates of the point \( X \) are \( (\sqrt{3}, 0) \), then the value of \( XP \cdot XQ \) is:

• A. \( \frac{4(2 + \sqrt{3})}{3} \)
• B. \( \frac{4(2 - \sqrt{3})}{2} \)
• C. \( \frac{5(2 + \sqrt{3})}{3} \)
• D. \( \frac{5(2 - \sqrt{3})}{3} \)

Hint:

When dealing with distances between points and intersections of curves, forming a quadratic equation in one variable often simplifies the problem using Vieta's formulas.

Answer 1

The Correct Option is A

Step 1: Find the coordinates of the intersection points \( P \) and \( Q \).
The equation of the line is \( y = \sqrt{3}x - 3 \).
The equation of the parabola is \( y^2 = x + 2 \).
Substitute the expression for \( y \) from the line equation into the parabola equation:
\[ (\sqrt{3}x - 3)^2 = x + 2 \] \[ 3x^2 - 6\sqrt{3}x + 9 = x + 2 \] \[ 3x^2 - (6\sqrt{3} + 1)x + 7 = 0 \] This is a quadratic equation in \( x \). Let the roots of this equation be \( x_1 \) and \( x_2 \), which are the x-coordinates of the points \( P \) and \( Q \). Using Vieta's formulas, we have: \[ x_1 + x_2 = \frac{6\sqrt{3} + 1}{3} \] \[ x_1 x_2 = \frac{7}{3} \] The corresponding y-coordinates are \( y_1 = \sqrt{3}x_1 - 3 \) and \( y_2 = \sqrt{3}x_2 - 3 \).
So, the points \( P \) and \( Q \) are \( (x_1, \sqrt{3}x_1 - 3) \) and \( (x_2, \sqrt{3}x_2 - 3) \).

Step 2: Calculate the distances \( XP \) and \( XQ \).
The coordinates of point \( X \) are \( (\sqrt{3}, 0) \). \[ XP^2 = (x_1 - \sqrt{3})^2 + (\sqrt{3}x_1 - 3 - 0)^2 = (x_1 - \sqrt{3})^2 + (\sqrt{3}(x_1 - \sqrt{3}))^2 \] \[ XP^2 = (x_1 - \sqrt{3})^2 + 3(x_1 - \sqrt{3})^2 = 4(x_1 - \sqrt{3})^2 \] \[ XP = 2|x_1 - \sqrt{3}| \] Similarly, \[ XQ^2 = (x_2 - \sqrt{3})^2 + (\sqrt{3}x_2 - 3 - 0)^2 = (x_2 - \sqrt{3})^2 + (\sqrt{3}(x_2 - \sqrt{3}))^2 \] \[ XQ^2 = (x_2 - \sqrt{3})^2 + 3(x_2 - \sqrt{3})^2 = 4(x_2 - \sqrt{3})^2 \] \[ XQ = 2|x_2 - \sqrt{3}| \]
Step 3: Calculate the product \( XP \cdot XQ \).
\[ XP \cdot XQ = 4 |(x_1 - \sqrt{3})(x_2 - \sqrt{3})| = 4 |x_1 x_2 - \sqrt{3}(x_1 + x_2) + 3| \] Substitute the values of \( x_1 + x_2 \) and \( x_1 x_2 \) from Step 1: \[ XP \cdot XQ = 4 \left| \frac{7}{3} - \sqrt{3}\left(\frac{6\sqrt{3} + 1}{3}\right) + 3 \right| \] \[ XP \cdot XQ = 4 \left| \frac{7}{3} - \frac{18 + \sqrt{3}}{3} + \frac{9}{3} \right| \] \[ XP \cdot XQ = 4 \left| \frac{7 - 18 - \sqrt{3} + 9}{3} \right| \] \[ XP \cdot XQ = 4 \left| \frac{-2 - \sqrt{3}}{3} \right| = 4 \left( \frac{2 + \sqrt{3}}{3} \right) = \frac{4(2 + \sqrt{3})}{3} \]