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

Answer: (D)

$\left( \frac{1}{2}, -20 \right)$

Answer 2

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 $Q$.