Valid Parameters for Midpoint and Equation of Parabola Chord

To determine the valid parameters for the midpoint $(h, k)$ and the equation of the chord $2x + y = p$ of the parabola $y^2 = 16x$, we use the standard equation of a chord with a given midpoint $(h, k)$, which is given by $T = S_1$.

For the parabola $y^2 = 16x$:

• $T = y k - 8(x + h)$
• $S_1 = k^2 - 16h$

Setting $T = S_1$: $y k - 8x - 8h = k^2 - 16h$ $8x - ky + k^2 - 8h = 0 \quad \text{--- (Equation 1)}$

The given equation of the chord is: $2x + y = p \implies 2x + y - p = 0$

Multiplying this equation by $4$ gives: $8x + 4y - 4p = 0 \quad \text{--- (Equation 2)}$

Comparing the coefficients of Equation 1 and Equation 2:

1. Comparing the coefficient of $y$: $-k = 4 \implies k = -4$

2. Comparing the constant terms: $k^2 - 8h = -4p$

Substitute $k = -4$ into the constant term relation: $(-4)^2 - 8h = -4p$ $16 - 8h = -4p$ Dividing by $-4$: $p = 2h - 4$

1. For $(h, k)$ to be the midpoint of a real chord intersecting the parabola in two distinct points, the point $(h, k)$ must lie inside the parabola, which requires $S_1 < 0$: $k^2 - 16h < 0$ $(-4)^2 - 16h < 0 \implies 16 < 16h \implies h > 1$

Now, let's evaluate the given options:

• Option [A]: $p = -2, h = 2, k = -4$

  • Here $k = -4$ and $h = 2 > 1$.
  • $p = 2h - 4 = 2(2) - 4 = 0 \neq -2$. Hence, this option is incorrect.

• Option [B]: $p = -1, h = 1, k = -3$

  • Here $k = -3 \neq -4$. Hence, this option is incorrect.

• Option [C]: $p = 2, h = 3, k = -4$

  • Here $k = -4$ and $h = 3 > 1$.
  • $p = 2(3) - 4 = 2$, which matches the given $p = 2$. Hence, this option is correct.

• Option [D]: $p = 5, h = 4, k = -3$

  • Here $k = -3 \neq -4$. Hence, this option is incorrect.

Thus, the correct option is C.