If `(2,-8)` is at an end of a focal chord of the parabola `y^2 = 32x`, then the other end of the chórd is

Options

A. `(-2,8)`
B. `(32,-32)`
C. `(32,32)`
D. none

Answer

To find the other end of the focal chord of the parabola given by the equation \( y^2 = 32x \), we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given equation of the parabola is \( y^2 = 32x \). We can compare this with the standard form \( y^2 = 4ax \) to find the value of \( a \).

\[ 4a = 32 \implies a = 8 \]

**Hint:** The value of \( a \) represents the distance from the vertex to the focus of the parabola.

### Step 2: Use the coordinates of the given point
The point \( (2, -8) \) is given as one end of the focal chord. We denote this point as \( A(t_1) \) where:

\[ A(t_1) = (at_1^2, 2at_1) = (8t_1^2, 16t_1) \]

We know that \( A(t_1) = (2, -8) \).

**Hint:** The coordinates of any point on the parabola can be expressed in terms of the parameter \( t \).

### Step 3: Set up equations based on the coordinates
From the x-coordinate:

\[ 8t_1^2 = 2 \implies t_1^2 = \frac{2}{8} = \frac{1}{4} \implies t_1 = \pm \frac{1}{2} \]

From the y-coordinate:

\[ 16t_1 = -8 \implies t_1 = -\frac{8}{16} = -\frac{1}{2} \]

**Hint:** The parameter \( t_1 \) can be either positive or negative, but both will yield valid points on the parabola.

### Step 4: Find the other end of the focal chord
For the other end of the focal chord, we need to find \( t_2 \). The relationship for focal chords is that \( t_1 \cdot t_2 = -1 \).

Given \( t_1 = -\frac{1}{2} \):

\[ t_2 = -\frac{1}{t_1} = -\frac{1}{-\frac{1}{2}} = 2 \]

**Hint:** The product of the parameters of the endpoints of a focal chord is always \(-1\).

### Step 5: Calculate the coordinates of the other end
Now we can find the coordinates of the point \( B(t_2) \):

\[ B(t_2) = (8t_2^2, 16t_2) = (8 \cdot 2^2, 16 \cdot 2) = (8 \cdot 4, 32) = (32, 32) \]

**Hint:** Substitute the value of \( t_2 \) into the parametric equations to find the coordinates.

### Conclusion
Thus, the other end of the focal chord is \( (32, 32) \).