Let P be the point (1, 0) and Q a point on the parabola $y^{2}=8x$. The locus of midpoint of PQ is:

Question

Accepted Answer

To determine the locus of the midpoint of the line segment PQ, where P is the point (1, 0) and Q is a point on the parabola $ y^2 = 8x $, we need to find the coordinates of the midpoint M of PQ and then derive its locus.

1. **Coordinates of Q:**
   Let the coordinates of Q be $(x_1, y_1)$ such that $y_1^2 = 8x_1$.

2. **Midpoint M of PQ:**
   The coordinates of the midpoint M of PQ are given by:
   $$
   M = \left( \frac{1 + x_1}{2}, \frac{0 + y_1}{2} \right) = \left( \frac{1 + x_1}{2}, \frac{y_1}{2} \right)
   $$

3. **Expressing $x_1$ and $y_1$ in terms of M:**
   Let the coordinates of M be $(h, k)$. Then:
   $$
   h = \frac{1 + x_1}{2} \quad \Rightarrow \quad x_1 = 2h - 1
   $$
   $$
   k = \frac{y_1}{2} \quad \Rightarrow \quad y_1 = 2k
   $$

4. **Substitute into the parabola equation:**
   Since $y_1^2 = 8x_1$, substitute $y_1 = 2k$ and $x_1 = 2h - 1$:
   $$
   (2k)^2 = 8(2h - 1)
   $$
   $$
   4k^2 = 16h - 8
   $$
   $$
   k^2 = 4h - 2
   $$

5. **Equation of the locus:**
   The equation $k^2 = 4h - 2$ represents the locus of the midpoint M, which can be rewritten as:
   $$
   y^2 = 4x - 2
   $$

This equation matches Option 4: $y^2 - 4x + 2 = 0$.

**Explanation for Incorrect Options:**

- **Option 1: $x^2 - 4y + 2 = 0$**
  - This equation suggests a different relationship between x and y that does not match the derived equation for the locus of M.

- **Option 2: $x^2 + 4y + 2 = 0$**
  - This equation implies a different form and relationship, which does not correspond to the derived equation $y^2 = 4x - 2$.

- **Option 3: $y^2 + 4x + 2 = 0$**
  - This equation suggests a different form of a parabola or conic section that does not match the derived equation for the locus of M.

Thus, the correct answer is Option 4: $y^2 - 4x + 2 = 0$, which accurately represents the locus of the midpoint of PQ.