I : The length of the latus rectum of the parabola `y^(2)+8x-2y+17=0` is 8 .
II: The focal distance of the point (9,6) on the parabola `y^(2)=4x` is 10

Options

A. 2
B. 4
C. 8 , 10
D. 16

Answer

To solve the given problem, we will break it down into two parts as indicated in the question.

### Part I: Length of the Latus Rectum of the Parabola

1. **Given Equation**: The equation of the parabola is \( y^2 + 8x - 2y + 17 = 0 \).

2. **Rearranging the Equation**: We need to rearrange this equation into the standard form of a parabola. Start by isolating the \( y \) terms:
\[ y^2 - 2y + 8x + 17 = 0 \]

3. **Completing the Square**: We complete the square for the \( y \) terms:
\[ y^2 - 2y = (y - 1)^2 - 1 \]
Substitute this back into the equation:
\[ (y - 1)^2 - 1 + 8x + 17 = 0 \]
Simplifying gives:
\[ (y - 1)^2 + 8x + 16 = 0 \]
Rearranging this, we get:
\[ (y - 1)^2 = -8(x + 2) \]

4. **Identifying Parameters**: Now, we can compare this with the standard form \( (y - k)^2 = -4a(x - h) \). Here, we have:
- \( k = 1 \)
- \( h = -2 \)
- \( 4a = 8 \) which implies \( a = 2 \).

5. **Finding the Length of the Latus Rectum**: The length of the latus rectum \( L \) is given by the formula:
\[ L = 4a \]
Substituting \( a = 2 \):
\[ L = 4 \times 2 = 8 \]

### Part II: Focal Distance of the Point (9, 6) on the Parabola \( y^2 = 4x \)

1. **Given Equation**: The equation of the parabola is \( y^2 = 4x \).

2. **Identifying Parameters**: This is already in the standard form \( y^2 = 4ax \). Here, we have:
- \( 4a = 4 \) which implies \( a = 1 \).
- The focus of the parabola is at \( (a, 0) = (1, 0) \).

3. **Finding the Focal Distance**: The focal distance \( SP \) from the point \( P(9, 6) \) to the focus \( S(1, 0) \) can be calculated using the distance formula:
\[ SP = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (1, 0) \) and \( (x_2, y_2) = (9, 6) \):
\[ SP = \sqrt{(9 - 1)^2 + (6 - 0)^2} \]
Calculating gives:
\[ SP = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Summary of Results
- The length of the latus rectum of the parabola \( y^2 + 8x - 2y + 17 = 0 \) is **8**.
- The focal distance of the point \( (9, 6) \) on the parabola \( y^2 = 4x \) is **10**.

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