Find the locus of the middle points of the chords of the circle `x^(2) + y^( 2) = 4 ( y + 1)` drawn through the origin.

Answer

To find the locus of the midpoints of the chords of the circle given by the equation \( x^2 + y^2 = 4(y + 1) \) that are drawn through the origin, we can follow these steps:

### Step 1: Rewrite the Circle's Equation
First, we rewrite the equation of the circle in standard form. The given equation is:

\[ x^2 + y^2 = 4(y + 1) \]

Expanding this, we have:

\[ x^2 + y^2 = 4y + 4 \]

Rearranging it gives:

\[ x^2 + y^2 - 4y - 4 = 0 \]

### Step 2: Complete the Square
Next, we complete the square for the \( y \) terms:

\[ x^2 + (y^2 - 4y) - 4 = 0 \]

Completing the square for \( y^2 - 4y \):

\[ y^2 - 4y = (y - 2)^2 - 4 \]

Substituting back, we get:

\[ x^2 + (y - 2)^2 - 4 - 4 = 0 \]

This simplifies to:

\[ x^2 + (y - 2)^2 = 8 \]

This represents a circle centered at \( (0, 2) \) with a radius of \( 2\sqrt{2} \).

### Step 3: Equation of the Chord
Let the midpoint of the chord be \( M(h, k) \). The equation of the chord with midpoint \( (h, k) \) that passes through the origin can be expressed using the formula:

\[ hx + ky = h^2 + k^2 - 4k \]

### Step 4: Substitute the Origin into the Chord Equation
Since the chord passes through the origin \( (0, 0) \), we substitute \( x = 0 \) and \( y = 0 \) into the chord equation:

\[ h(0) + k(0) = h^2 + k^2 - 4k \]

This simplifies to:

\[ 0 = h^2 + k^2 - 4k \]

### Step 5: Rearranging the Equation
Rearranging gives us:

\[ h^2 + k^2 - 4k = 0 \]

### Step 6: Substitute \( h \) and \( k \) with \( x \) and \( y \)
Now, replace \( h \) and \( k \) with \( x \) and \( y \):

\[ x^2 + y^2 - 4y = 0 \]

### Step 7: Final Form
This can be rearranged to:

\[ x^2 + (y^2 - 4y) = 0 \]

Completing the square for \( y^2 - 4y \):

\[ x^2 + (y - 2)^2 - 4 = 0 \]

Thus, we have:

\[ x^2 + (y - 2)^2 = 4 \]

### Conclusion
The locus of the midpoints of the chords of the circle drawn through the origin is:

\[ x^2 + (y - 2)^2 = 4 \]

This represents a circle centered at \( (0, 2) \) with a radius of \( 2 \).