Find the equation to the locus of the point, the square of whose distance from origin is 4 times its y-coordinate.

Answer

To find the equation of the locus of the point whose distance from the origin squared is 4 times its y-coordinate, we can follow these steps:

### Step 1: Define the point
Let the point be \( P(x, y) \).

### Step 2: Calculate the distance from the origin
The distance \( d \) of the point \( P(x, y) \) from the origin \( O(0, 0) \) is given by the distance formula:
\[ d = \sqrt{x^2 + y^2} \]

### Step 3: Square the distance
Now, we square the distance:
\[ d^2 = x^2 + y^2 \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the square of the distance from the origin is equal to 4 times the y-coordinate:
\[ d^2 = 4y \]

### Step 5: Substitute the squared distance into the equation
Substituting the expression for \( d^2 \) from Step 3 into the equation from Step 4 gives us:
\[ x^2 + y^2 = 4y \]

### Step 6: Rearrange the equation
To express this in standard form, we can rearrange the equation:
\[ x^2 + y^2 - 4y = 0 \]

### Final Equation
Thus, the equation of the locus is:
\[ x^2 + y^2 - 4y = 0 \]

---