The focal distance of a point on a parabola `y^2=8x` is 8. Find it .

Answer

To find the point on the parabola \( y^2 = 8x \) where the focal distance is 8, we will follow these steps:

### Step 1: Understand the Parabola and Focal Distance
The given parabola is \( y^2 = 8x \). The focus of this parabola is at the point \( (2, 0) \). The focal distance of a point \( (x, y) \) on the parabola is the distance from that point to the focus.

### Step 2: Write the Distance Formula
The distance \( d \) from a point \( (x, y) \) to the focus \( (2, 0) \) is given by:
\[ d = \sqrt{(x - 2)^2 + (y - 0)^2} \]
Since we know the focal distance is 8, we can set up the equation:
\[ \sqrt{(x - 2)^2 + y^2} = 8 \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides:
\[ (x - 2)^2 + y^2 = 64 \]

### Step 4: Substitute the Parabola Equation
From the equation of the parabola \( y^2 = 8x \), we can substitute \( y^2 \) in the distance equation:
\[ (x - 2)^2 + 8x = 64 \]

### Step 5: Expand and Rearrange
Now, we expand the left side:
\[ (x^2 - 4x + 4) + 8x = 64 \]
Combining like terms gives:
\[ x^2 + 4x + 4 = 64 \]
Now, rearranging this equation:
\[ x^2 + 4x + 4 - 64 = 0 \]
\[ x^2 + 4x - 60 = 0 \]

### Step 6: Factor the Quadratic Equation
Next, we will factor the quadratic equation:
\[ (x - 6)(x + 10) = 0 \]
Setting each factor to zero gives us:
\[ x - 6 = 0 \quad \text{or} \quad x + 10 = 0 \]
Thus, we find:
\[ x = 6 \quad \text{or} \quad x = -10 \]

### Step 7: Find Corresponding y-values
Now we will find the corresponding \( y \)-values for each \( x \):
1. For \( x = 6 \):
\[ y^2 = 8(6) = 48 \quad \Rightarrow \quad y = \pm \sqrt{48} = \pm 4\sqrt{3} \]
So, the points are \( (6, 4\sqrt{3}) \) and \( (6, -4\sqrt{3}) \).

2. For \( x = -10 \):
\[ y^2 = 8(-10) \quad \text{(not valid since } y^2 \text{ cannot be negative)} \]

### Conclusion
The valid points on the parabola where the focal distance is 8 are:
\[ (6, 4\sqrt{3}) \quad \text{and} \quad (6, -4\sqrt{3}) \]