The point lying on $y^{2}=8x$ and at a minimum distance from $x^{2}+(y+6)^{2}=1$ will be:

Question

Accepted Answer

To determine the point on the parabola $ y^2 = 8x $ that is at a minimum distance from the circle $ x^2 + (y+6)^2 = 1 $, we need to analyze the geometry of the situation.

### Explanation for Option 1: (2, -4)

1. **Equation of the Parabola**: The given parabola is $ y^2 = 8x $. This is a standard parabola that opens to the right with its vertex at the origin (0,0).

2. **Equation of the Circle**: The circle is given by $ x^2 + (y+6)^2 = 1 $. This circle is centered at (0, -6) with a radius of 1.

3. **Minimum Distance**: To find the point on the parabola that is closest to the circle, we need to minimize the distance between a point $(x_1, y_1)$ on the parabola and the center of the circle (0, -6).

4. **Distance Formula**: The distance $d$ between a point $(x_1, y_1)$ and the center of the circle is given by:
   $$
   d = \sqrt{(x_1 - 0)^2 + (y_1 + 6)^2}
   $$
   Since $y_1^2 = 8x_1$, we substitute $x_1 = \frac{y_1^2}{8}$ into the distance formula:
   $$
   d = \sqrt{\left(\frac{y_1^2}{8}\right)^2 + (y_1 + 6)^2}
   $$

5. **Checking Option 1**: For the point (2, -4):
   - Verify it lies on the parabola: $(-4)^2 = 16$ and $8 \times 2 = 16$, so it satisfies $y^2 = 8x$.
   - Calculate the distance to the circle's center: 
     $$
     d = \sqrt{(2 - 0)^2 + (-4 + 6)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
     $$
   This is the minimum distance achievable given the constraints.

### Explanation for Other Options:

- **Option 2: (2, 4)**
  - This point does not satisfy the parabola equation $y^2 = 8x$ because $4^2 = 16$ and $8 \times 2 = 16$, but the distance to the circle's center is greater than the minimum found in Option 1.

- **Option 3: (18, -12)**
  - This point does not satisfy the parabola equation $y^2 = 8x$ because $(-12)^2 = 144$ and $8 \times 18 = 144$, but the distance to the circle's center is much larger than the minimum found in Option 1.

- **Option 4: (8, 8)**
  - This point does not satisfy the parabola equation $y^2 = 8x$ because $8^2 = 64$ and $8 \times 8 = 64$, but the distance to the circle's center is significantly larger than the minimum found in Option 1.

In conclusion, Option 1 (2, -4) is the correct answer as it satisfies the parabola equation and provides the minimum distance to the circle's center.