The locus of a point which divides the line segment joining the point (0, -1) and a point on the parabola,`x^(2) = 4y` internally in the ratio 1: 2, is:

Options

A. `9x^(2) - 12y = 8`
B. `4x^(2) - 3y = 8`
C. `x^(2) - 3y = 2`
D. `9x^(2) - 3y =2`

Answer

To find the locus of a point that divides the line segment joining the point \( (0, -1) \) and a point on the parabola \( x^2 = 4y \) internally in the ratio \( 1:2 \), we can follow these steps:

### Step 1: Identify the points
Let the point on the parabola be \( (x_1, y_1) \). Since the point lies on the parabola \( x^2 = 4y \), we can express \( y_1 \) in terms of \( x_1 \):
\[ y_1 = \frac{x_1^2}{4} \]

### Step 2: Use the section formula
The coordinates of the point \( (h, k) \) that divides the line segment joining \( (0, -1) \) and \( (x_1, y_1) \) in the ratio \( 1:2 \) can be found using the section formula:
\[ h = \frac{m_2 x_1 + m_1 x_2}{m_1 + m_2} \quad \text{and} \quad k = \frac{m_2 y_1 + m_1 y_2}{m_1 + m_2} \]
where \( m_1 = 1 \), \( m_2 = 2 \), \( (x_2, y_2) = (0, -1) \).

Substituting the values:
\[ h = \frac{2 \cdot 0 + 1 \cdot x_1}{1 + 2} = \frac{x_1}{3} \]
\[ k = \frac{2 \cdot (-1) + 1 \cdot y_1}{1 + 2} = \frac{-2 + y_1}{3} \]

### Step 3: Substitute \( y_1 \)
Now, substitute \( y_1 = \frac{x_1^2}{4} \) into the equation for \( k \):
\[ k = \frac{-2 + \frac{x_1^2}{4}}{3} \]

### Step 4: Rearranging the equation
Multiply through by 3 to eliminate the denominator:
\[ 3k = -2 + \frac{x_1^2}{4} \]
Rearranging gives:
\[ \frac{x_1^2}{4} = 3k + 2 \]
Multiplying both sides by 4:
\[ x_1^2 = 12k + 8 \]

### Step 5: Express \( h \) in terms of \( k \)
From \( h = \frac{x_1}{3} \), we have:
\[ x_1 = 3h \]
Substituting this into the equation \( x_1^2 = 12k + 8 \):
\[ (3h)^2 = 12k + 8 \]
\[ 9h^2 = 12k + 8 \]

### Step 6: Rearranging to find the locus
Rearranging gives us:
\[ 9h^2 - 12k - 8 = 0 \]

### Step 7: Final form of the locus
This can be expressed as:
\[ 3h^2 - 4k - \frac{8}{3} = 0 \]
or in a more standard form:
\[ 3x^2 - 4y - \frac{8}{3} = 0 \]

### Conclusion
The locus of the point that divides the line segment in the given ratio is:
\[ 3x^2 - 4y - \frac{8}{3} = 0 \]