If O is the origin and Q is a variable, point on `x^(2)=4y`, find the locus of the mid-point of OQ.

Answer

To find the locus of the midpoint of the line segment joining the origin \( O(0, 0) \) and a variable point \( Q(a, b) \) on the parabola defined by the equation \( x^2 = 4y \), we can follow these steps:

### Step 1: Identify the coordinates of the points
- The origin \( O \) has coordinates \( (0, 0) \).
- The variable point \( Q \) has coordinates \( (a, b) \), where \( Q \) lies on the parabola \( x^2 = 4y \).

### Step 2: Use the midpoint formula
The formula for the midpoint \( M \) of a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
For points \( O(0, 0) \) and \( Q(a, b) \), the midpoint \( M \) is:
\[ M = \left( \frac{0 + a}{2}, \frac{0 + b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right) \]

### Step 3: Set the coordinates of the midpoint
Let the coordinates of the midpoint \( M \) be \( (h, k) \). Thus, we have:
\[ h = \frac{a}{2} \quad \text{and} \quad k = \frac{b}{2} \]
From these equations, we can express \( a \) and \( b \) in terms of \( h \) and \( k \):
\[ a = 2h \quad \text{and} \quad b = 2k \]

### Step 4: Substitute into the parabola equation
Since point \( Q(a, b) \) lies on the parabola \( x^2 = 4y \), we substitute \( a \) and \( b \) into this equation:
\[ (2h)^2 = 4(2k) \]
This simplifies to:
\[ 4h^2 = 8k \]

### Step 5: Rearrange the equation
Dividing both sides by 4 gives:
\[ h^2 = 2k \]

### Step 6: Write the locus in standard form
In terms of \( h \) and \( k \), we can rewrite this as:
\[ h^2 = 2k \]
This represents the locus of the midpoint \( M \) in the form of a parabola.

### Final Answer
The locus of the midpoint of \( OQ \) is given by:
\[ x^2 = 2y \]

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