To find the locus of point Q, which divides the segment joining point A (0, -1) and point P on the parabola \(x^2 = 4y\) in the ratio 1:2, we can follow these steps:
### Step 1: Identify the coordinates of point P
Since point P lies on the parabola \(x^2 = 4y\), we can express the coordinates of point P in terms of a parameter \(t\). The parametric equations for the parabola are:
\[
P(2t, t^2)
\]
### Step 2: Use the section formula to find the coordinates of point Q
Point Q divides the segment joining A (0, -1) and P (2t, t^2) in the ratio 1:2. According to the section formula, if a point Q divides the segment joining points A \((x_1, y_1)\) and B \((x_2, y_2)\) in the ratio \(m:n\), then the coordinates of point Q are given by:
\[
Q\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Here, \(m = 1\), \(n = 2\), \(A(0, -1)\), and \(P(2t, t^2)\).
Substituting the values:
\[
Q\left(\frac{1 \cdot 2t + 2 \cdot 0}{1 + 2}, \frac{1 \cdot t^2 + 2 \cdot (-1)}{1 + 2}\right) = Q\left(\frac{2t}{3}, \frac{t^2 - 2}{3}\right)
\]
### Step 3: Set the coordinates of Q in terms of x and y
Let the coordinates of Q be:
\[
Q\left(x, y\right) \Rightarrow x = \frac{2t}{3} \quad \text{and} \quad y = \frac{t^2 - 2}{3}
\]
### Step 4: Express \(t\) in terms of \(x\)
From the equation \(x = \frac{2t}{3}\), we can express \(t\) as:
\[
t = \frac{3x}{2}
\]
### Step 5: Substitute \(t\) into the equation for \(y\)
Now substitute \(t\) into the equation for \(y\):
\[
y = \frac{t^2 - 2}{3} = \frac{\left(\frac{3x}{2}\right)^2 - 2}{3}
\]
Calculating \(t^2\):
\[
t^2 = \left(\frac{3x}{2}\right)^2 = \frac{9x^2}{4}
\]
Now substituting this back into the equation for \(y\):
\[
y = \frac{\frac{9x^2}{4} - 2}{3} = \frac{9x^2 - 8}{12}
\]
### Step 6: Rearranging the equation
Multiplying both sides by 12 to eliminate the fraction:
\[
12y = 9x^2 - 8
\]
Rearranging gives us the final equation:
\[
9x^2 = 12y + 8
\]
### Final Result
Thus, the locus of point Q is given by:
\[
9x^2 = 12y + 8
\]
