Question

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 :

• A. $9x^2 - 12y = 8$
• B. $4x^2 - 3y = 2$
• C. $x^2 - 3y = 2$
• D. $9x^2 -3y = 2$

Answer 1

The Correct Option is A

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, follow these steps:

1. Let the coordinates of any point on the parabola be \((x_1, y_1)\). Since the point is on the parabola \(x^2 = 4y\), we have \(x_1^2 = 4y_1\).
2. According to the section formula, the coordinates of the point dividing the line segment joining \((0, -1)\) and \((x_1, y_1)\) in the ratio 1:2 are given by:

\[ \left( \frac{1 \cdot x_1 + 2 \cdot 0}{1 + 2}, \frac{1 \cdot y_1 + 2 \cdot (-1)}{1 + 2} \right) = \left( \frac{x_1}{3}, \frac{y_1 - 2}{3} \right) \]

1. Let this point be \((h, k)\). Therefore, we can write:

\[ h = \frac{x_1}{3} \quad \text{and} \quad k = \frac{y_1 - 2}{3} \]

1. From the first equation, solving for \(x_1\), we get:

\[ x_1 = 3h \]

1. From the second equation, solving for \(y_1\), we get:

\[ y_1 = 3k + 2 \]

1. Since \((x_1, y_1)\) lies on the parabola \(x_1^2 = 4y_1\), substituting the expressions for \(x_1\) and \(y_1\) we get:

\[ (3h)^2 = 4(3k + 2) \]

1. Expanding and simplifying this equation:

\[ 9h^2 = 12k + 8 \]

1. Rearranging terms, the equation becomes:

\[ 9h^2 - 12k = 8 \]

1. Thus, the locus of the point is represented by the equation: \[ 9x^2 - 12y = 8 \]

Therefore, the correct answer is \(9x^2 - 12y = 8\), which matches option (A).