If a chord, which is not a tangent, of the parabola `y^(2)=16x` has the equation 2x+y=p, and midpoint (h,k), then which of the following is (are) possible value(s) of p,h and k ?

Options

A. p=5, h=4, k=-3
B. p=-1, h=1, k=-3
C. p=-2, h=2, k=-4
D. p=2, h=3, k=-4

Explanation

To solve the problem, we need to find the possible values of \( p \), \( h \), and \( k \) given the conditions of the chord of the parabola \( y^2 = 16x \) and the equation of the chord \( 2x + y = p \).

### Step-by-step Solution:

1. **Identify the Parabola and Chord Equation**:
The equation of the parabola is given as:
\[ y^2 = 16x \]
The equation of the chord is given as:
\[ 2x + y = p \]

2. **Assume the Midpoint**:
Let the midpoint of the chord be \( (h, k) \).

3. **Equation of the Chord in Terms of Midpoint**:
The equation of the chord passing through the midpoint can be expressed using the parametric form of the parabola. The equation can be derived as:
\[ y(k - 8) = 16(h - x) \]
Rearranging gives:
\[ ky - 8x = k^2 - 16h \]

4. **Comparing the Two Chord Equations**:
We have two equations:
- From the chord equation: \( 2x + y = p \)
- From the midpoint equation: \( ky - 8x = k^2 - 16h \)

We can rewrite the chord equation as:
\[ y = p - 2x \]
Substituting this into the midpoint equation:
\[ k(p - 2x) - 8x = k^2 - 16h \]
Simplifying gives:
\[ kp - 2kx - 8x = k^2 - 16h \]
Rearranging terms results in:
\[ (8 + 2k)x = kp - k^2 + 16h \]

5. **Setting Conditions for a Chord**:
For the line to be a chord and not a tangent, the discriminant of the resulting quadratic equation must be positive. This leads to the conditions:
\[ \frac{k}{1} = \frac{-8}{2} = \frac{k^2 - 8h}{p} \]

6. **Finding Relations**:
From the above equations, we can derive:
\[ k = -4 \quad \text{(since } k = -8/2 \text{)} \]
Substituting \( k = -4 \) into the equation:
\[ -4^2 - 8h = -4p \]
This simplifies to:
\[ 16 - 8h = -4p \]
Rearranging gives:
\[ 4p = 8h - 16 \quad \Rightarrow \quad p = 2h - 4 \]

7. **Finding Possible Values**:
Now we can test possible integer values for \( h \):
- If \( h = 2 \):
\[ p = 2(2) - 4 = 0 \]
- If \( h = 3 \):
\[ p = 2(3) - 4 = 2 \]

Thus, we have two sets of values:
- For \( h = 2 \), \( p = 0 \), \( k = -4 \)
- For \( h = 3 \), \( p = 2 \), \( k = -4 \)

### Conclusion:
The possible values of \( (p, h, k) \) are:
1. \( (0, 2, -4) \)
2. \( (2, 3, -4) \)