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

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the parabola and its parameters
The given parabola is \( y^2 = 16x \). We can compare this with the standard form of a parabola \( y^2 = 4ax \). Here, we have:
\[ 4a = 16 \implies a = 4 \]

### Step 2: Write the equation of the chord
The equation of the chord of a parabola can be expressed as \( S' = T \). For the parabola \( y^2 = 4ax \), this translates to:
\[ y_1^2 - 4ax_1 = y y_1 - 2ax - x_1 \]
Substituting \( a = 4 \), we have:
\[ y_1^2 - 16x_1 = yy_1 - 8x - x_1 \]

### Step 3: Substitute the midpoint values
Let the midpoint of the chord be \( (h, k) \). Thus, substituting \( y_1 = k \) and \( x_1 = h \) into the equation gives:
\[ k^2 - 16h = ky - 8x - h \]

### Step 4: Rearranging the equation
Rearranging the equation leads to:
\[ k^2 - 16h + h = ky - 8x \]
This simplifies to:
\[ k^2 - 15h = ky - 8x \]

### Step 5: Compare with the given chord equation
The equation of the chord is given as \( 2x + y = p \). We can rewrite this as:
\[ y = p - 2x \]
Now, substituting \( y \) into the rearranged equation gives:
\[ k^2 - 15h = k(p - 2x) - 8x \]
Expanding this, we have:
\[ k^2 - 15h = kp - 2kx - 8x \]
This can be rearranged to:
\[ k^2 - 15h = kp - (2k + 8)x \]

### Step 6: Coefficients comparison
Since both equations represent the same chord, the coefficients of \( x \) and the constant terms must be equal:
1. Coefficient of \( x \):
\[ 2k + 8 = 0 \implies k = -4 \]
2. Constant term:
\[ k^2 - 15h = kp \implies (-4)^2 - 15h = -4p \]
Simplifying gives:
\[ 16 - 15h = -4p \implies 4p = 15h - 16 \implies p = \frac{15h - 16}{4} \]

### Step 7: Finding possible values of \( p, h, k \)
Now we have:
- \( k = -4 \)
- \( p = \frac{15h - 16}{4} \)

### Step 8: Check the options
Now we can check the given options for possible values of \( p, h, k \).

1. For option 1: \( p = 2, h = 3, k = -4 \)
\[ 2 = \frac{15(3) - 16}{4} \implies 2 = \frac{45 - 16}{4} \implies 2 = \frac{29}{4} \text{ (not valid)} \]

2. For option 2: \( p = 4, h = 4, k = -4 \)
\[ 4 = \frac{15(4) - 16}{4} \implies 4 = \frac{60 - 16}{4} \implies 4 = \frac{44}{4} \text{ (valid)} \]

3. For option 3: \( p = 6, h = 5, k = -4 \)
\[ 6 = \frac{15(5) - 16}{4} \implies 6 = \frac{75 - 16}{4} \implies 6 = \frac{59}{4} \text{ (not valid)} \]

4. For option 4: \( p = 8, h = 6, k = -4 \)
\[ 8 = \frac{15(6) - 16}{4} \implies 8 = \frac{90 - 16}{4} \implies 8 = \frac{74}{4} \text{ (not valid)} \]

### Conclusion
The only valid option is option 2, where \( p = 4, h = 4, k = -4 \).