let `P` be the point `(1, 0)` and `Q` be a point on the locus `y^2= 8x`. The locus of the midpoint of `PQ` is

Options

A. `y^(2)+4x+2=0`
B. `y^(2)-4x+2=0`
C. `x^(2)-4y+2=0`
D. `x^(2)+4y+2=0`

Explanation

To find the locus of the midpoint of the segment joining the point \( P(1, 0) \) and a point \( Q \) on the parabola defined by \( y^2 = 8x \), we can follow these steps:

### Step 1: Define the Points
Let the point \( Q \) on the parabola \( y^2 = 8x \) be represented as \( Q(t) = (t^2/2, t) \), where \( t \) is a parameter that gives the coordinates of points on the parabola.

### Step 2: Find the Midpoint of \( PQ \)
The midpoint \( M \) of the segment \( PQ \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( P(1, 0) \) and \( Q(t) = (t^2/2, t) \):
\[ M = \left( \frac{1 + \frac{t^2}{2}}{2}, \frac{0 + t}{2} \right) = \left( \frac{2 + t^2}{4}, \frac{t}{2} \right) \]

### Step 3: Set Variables for Midpoint
Let \( h = \frac{2 + t^2}{4} \) and \( k = \frac{t}{2} \).

### Step 4: Express \( t \) in Terms of \( k \)
From \( k = \frac{t}{2} \), we can express \( t \) as:
\[ t = 2k \]

### Step 5: Substitute \( t \) Back into the Equation for \( h \)
Now substitute \( t = 2k \) into the equation for \( h \):
\[ h = \frac{2 + (2k)^2}{4} = \frac{2 + 4k^2}{4} = \frac{1 + 2k^2}{2} \]

### Step 6: Rearrange to Find the Locus
Now, we can rearrange the equation for \( h \):
\[ 2h = 1 + 2k^2 \]
This can be rearranged to:
\[ 2k^2 = 2h - 1 \]
\[ k^2 = h - \frac{1}{2} \]

### Step 7: Express in Standard Form
Substituting back the variables \( h \) and \( k \) gives us the locus of the midpoint:
\[ k^2 = h - \frac{1}{2} \]
This represents a parabola.

### Final Locus Equation
Thus, the locus of the midpoint of \( PQ \) is given by:
\[ y^2 = 2x - 1 \]