Problem 99 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The locus of a point which divides the line segment joining the point \((0,-1)\) and a point on the parabola, \(x^{2}=4 y\), internally in the ratio \(1: 2\), is: (a) \(9 x^{2}-12 y=8\) (b) \(9 x^{2}-3 y=2\) (c) \(x^{2}-3 y=2\) (d) \(4 x^{2}-3 y=2\)

Step-by-Step Solution

Answer

The locus is given by option (a): \(9x^2 - 12y = 8\).

Step 1: Understand the Problem

We need to find the locus of a point that divides a line segment joining \((0, -1)\) and a point on the parabola \(x^2 = 4y\) in the ratio 1:2. The locus is the path or curve traced by this dividing point as the point on the parabola moves.

Step 2: Parameterize the Parabola

Consider a point \((x_1, y_1)\) on the parabola \(x^2 = 4y\). So, \(x_1 = x\) and \(y_1 = \frac{x^2}{4}\). This represents every point on the parabola.

Step 3: Apply Section Formula

The point that divides the line segment joining \((0, -1)\) and \((x, \frac{x^2}{4})\) in the ratio 1:2 can be found using the section formula. If the coordinates of the dividing point are \((X, Y)\), then:\[ X = \frac{1 \cdot x + 2 \cdot 0}{1 + 2} = \frac{x}{3} \]\[ Y = \frac{1 \cdot \frac{x^2}{4} + 2 \cdot (-1)}{1 + 2} = \frac{x^2}{12} - \frac{2}{3} \]

Step 4: Find Locus Equation

To find the locus, express \(X\) and \(Y\) in terms of each other. We have:\[ X = \frac{x}{3} \Rightarrow x = 3X \]Substitute \(x = 3X\) into the equation for \(Y\):\[ Y = \frac{(3X)^2}{12} - \frac{2}{3} \]\[ Y = \frac{9X^2}{12} - \frac{2}{3} \]Simplify this to:\[ Y = \frac{3X^2}{4} - \frac{2}{3} \]

Step 5: Simplify and Compare Equations

Multiply the entire equation by 12 to eliminate fractions:\[ 12Y = 9X^2 - 8 \]Rearrange to match the form of the given options:\[ 9X^2 - 12Y = 8 \]

Step 6: Identify the Correct Option

The equation \(9x^2 - 12y = 8\) matches option (a). Therefore, the locus of the point is represented by this equation.

Key Concepts

Section Formula

The section formula is a powerful tool used in coordinate geometry to determine the coordinates of a point that divides a line segment in a given ratio. This formula can be applied in various scenarios, such as internally or externally dividing a segment.
When a point divides a line segment internally in the ratio \(m:n\), the formula for the coordinates of this point \((X, Y)\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

• \(X = \frac{mx_2 + nx_1}{m + n}\)
• \(Y = \frac{my_2 + ny_1}{m + n}\)

This formula provides a straightforward way to calculate the intermediate points on a line connecting two given points, which is essential in creating constructions and solving problems involving shapes and paths.

Parabola Equation

A parabola is a unique type of curve and can be described by equations like \(x^2 = 4y\), which is a standard form for a parabola oriented vertically upwards. This form highlights the relationship between the \(x\) and \(y\) coordinates on the parabola.
The vertex of the parabola, defined by this equation, lies at the origin \((0,0)\), and the symmetry of the parabola ensures every \(x\) value corresponds to a unique \(y\) value. Understanding these relations is crucial for accurately parameterizing points on the parabola as seen in the given problem.
Exploring the nature of parabolas helps in recognizing how they relate to conic sections and their intersections with lines, circles, or other curves, which is a fundamental aspect of analytical geometry.

Line Segment Division

Dividing a line segment is a key concept in geometry, involving finding points at specific ratios along the segment. In the context of analytical geometry, it helps in locating specific points that align with geometric constraints.
The division of a line segment can be visualized as placing a point at a specific internal or external ratio along the segment. This internal division was applied in the exercise, where a point divided the segment into the ratio 1:2.
Understanding line segment division is essential, not just for simple problems but also for more complex constructions and determining loci, leading to deeper insights into the nature of geometry and coordinate systems.