Question

Let \( PQ \) be a chord of the parabola \( y^2 = 12x \) and the midpoint of \( PQ \) be at \( (4, 1) \). Then, which of the following points lies on the line passing through the points \( P \) and \( Q \)?

• A. (3, -3)
• B. \( \left( \frac{3}{2}, -16 \right) \)
• C. (2, -9)
• D. \( \left( \frac{1}{2}, -20 \right) \)

Answer 1

The Correct Option is D

Let \( P \) and \( Q \) be points on the parabola \( y^2 = 12x \) with coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \), respectively. Since \( (4, 1) \) is the midpoint of \( PQ \), we have:

\[ \frac{x_1 + x_2}{2} = 4 \implies x_1 + x_2 = 8, \] \[ \frac{y_1 + y_2}{2} = 1 \implies y_1 + y_2 = 2. \]

Since \( P \) and \( Q \) lie on the parabola \( y^2 = 12x \), we have:

\[ y_1^2 = 12x_1 \quad \text{and} \quad y_2^2 = 12x_2. \]

The equation of the chord of a parabola with a given midpoint can be derived as:

\[ y(y_1 + y_2) = 2x + x_1 + x_2. \]

Substituting \( y_1 + y_2 = 2 \) and \( x_1 + x_2 = 8 \), we get:

\[ y \cdot 2 = 2x + 8, \] \[ \implies y = x - 4. \]

Now, we substitute each option to check which one satisfies the equation \( y = x - 4 \).

• For option (1), \( (3, -3) \): \[ -3 \neq 3 - 4. \]
• For option (2), \( \left(\frac{3}{2}, -16\right) \): \[ -16 \neq \frac{3}{2} - 4. \]
• For option (3), \( (2, -9) \): \[ -9 \neq 2 - 4. \]
• For option (4), \( \left(\frac{1}{2}, -20\right) \): \[ -20 = \frac{1}{2} - 4. \]

Thus, the point \( \left(\frac{1}{2}, -20\right) \) lies on the line passing through points \( P \) and \( Q \).