To find the other extremity of the focal chord of the parabola \( y^2 = 8x \) given one extremity at \( \left(\frac{1}{2}, -2\right) \), we can follow these steps:
### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 8x \). We can compare this with the standard form \( y^2 = 4ax \) to identify \( a \):
\[
4a = 8 \implies a = 2
\]
**Hint:** Remember that in the standard form of a parabola, \( y^2 = 4ax \), \( a \) represents the distance from the vertex to the focus.
### Step 2: Determine the coordinates of the given point
The coordinates of the given extremity of the focal chord are:
\[
P\left(\frac{1}{2}, -2\right)
\]
Here, \( x = \frac{1}{2} \) and \( y = -2 \).
**Hint:** Make sure to note down the coordinates clearly as they will be used to find the parameter \( t_1 \).
### Step 3: Find the parameter \( t_1 \)
For the parabola, the coordinates of any point can be expressed in terms of the parameter \( t \) as:
\[
(x, y) = (at^2, 2at)
\]
Substituting \( a = 2 \):
\[
(x, y) = (2t^2, 4t)
\]
From the point \( P\left(\frac{1}{2}, -2\right) \), we have:
\[
4t_1 = -2 \implies t_1 = -\frac{1}{2}
\]
**Hint:** Use the \( y \)-coordinate to find the value of \( t_1 \) since it relates directly to the parameterization of the parabola.
### Step 4: Find the parameter \( t_2 \) for the other extremity
The relationship between the parameters of the endpoints of a focal chord is given by:
\[
t_1 \cdot t_2 = -1
\]
Substituting \( t_1 = -\frac{1}{2} \):
\[
-\frac{1}{2} \cdot t_2 = -1 \implies t_2 = 2
\]
**Hint:** Remember that the product of the parameters of the endpoints of a focal chord is always \(-1\).
### Step 5: Calculate the coordinates of the other extremity \( Q \)
Using \( t_2 = 2 \) in the parameterization:
\[
Q = (2t_2^2, 4t_2) = (2 \cdot 2^2, 4 \cdot 2) = (2 \cdot 4, 8) = (8, 8)
\]
**Hint:** Substitute \( t_2 \) into the parameterization formula to find the coordinates of the other extremity of the focal chord.
### Conclusion
Thus, the coordinates of the other extremity of the focal chord are:
\[
\boxed{(8, 8)}
\]
