Problem 123 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{O}\) be the vertex and \(\mathrm{Q}\) be any point on the parabola, \(\mathrm{x}^{2}=8 \mathrm{y}\). If the point \(\mathrm{P}\) divides the line segment OQ internally in the ratio \(1: 3\), then locus of \(\mathrm{P}\) is : (a) \(y^{2}=2 x\) (b) \(x^{2}=2 y\) (c) \(x^{2}=y\) (d) \(y^{2}=x\)

Step-by-Step Solution

Answer

The locus of \( P \) is \( x^2 = 2y \) (option b).

Step 1: Understand the Parabola Equation

The given parabola is described by the equation \( x^2 = 8y \). From this, we can identify the vertex as \( O(0,0) \), which is at the origin.

Step 2: Parameterize the Point Q

Consider a point \( Q \) on the parabola. This point can be parameterized as \( Q(t, \frac{t^2}{8}) \) since it must satisfy the parabola equation \( x^2 = 8y \).

Step 3: Use Section Formula to Find Coordinates of P

The point \( P \) divides the line segment \( OQ \) internally in the ratio 1:3. Using the section formula, the coordinates of \( P \) are given by \( P = \left( \frac{1 \cdot t + 3 \cdot 0}{1+3}, \frac{1 \cdot \frac{t^2}{8} + 3 \cdot 0}{1+3} \right) = \left( \frac{t}{4}, \frac{t^2}{32} \right) \).

Step 4: Find Relation Between Coordinates of P

In \( P \), we have \( x = \frac{t}{4} \) and \( y = \frac{t^2}{32} \). Express \( t \) from \( x = \frac{t}{4} \) as \( t = 4x \). Substitute \( t \) in \( y \): \( y = \frac{(4x)^2}{32} = \frac{16x^2}{32} = \frac{x^2}{2} \).

Step 5: Write the Locus of P

The derived relationship \( y = \frac{x^2}{2} \) is the equation of the locus of \( P \). Therefore, the locus of \( P \) is described by the equation \( x^2 = 2y \).

Key Concepts

Parabola

A parabola is a U-shaped curve that is one of the core concepts in conic sections. It is defined as the set of all points that are equidistant from a fixed point called the focus and a line known as the directrix.
A standard form of the parabola's equation is either in the form of \( y^2 = 4ax \) or \( x^2 = 4ay \), depending on its orientation.

• For a vertically oriented parabola like \( x^2 = 8y \), the vertex is typically at the origin, \( (0, 0) \), and this is where the curve is the narrowest.
• The axis of symmetry is a vertical line through the vertex, allowing the parabola to be symmetrical.

Understanding parabolas is crucial as they often appear in problems involving motion, optics, and various optimization scenarios.

Locus

The term "locus" refers to the set of all points that satisfy a specific condition or a set of conditions.
In coordinate geometry, finding the locus is a common exercise where you'll derive an equation that represents all the positions a point can occupy given its path constraint.

• In the context of the problem, the locus pertains to the path followed by point \( P \) as \( Q \) moves along the parabola.
• By understanding the transformation and division of segments, we derive the locus as \( x^2 = 2y \).

This helps visualize dynamic systems where position changes according to set rules and conditions.

Section Formula

The section formula is a mathematical method used to find the coordinates of a point that divides a line segment into a certain ratio.
It is particularly useful in coordinate geometry.

• If point \( P \) divides a line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the internal ratio \( m:n \), then coordinates of \( P \) are given by:

\[ (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
This simple formula applies the concept of weighted averages to determine the location of dividing points on a segment. It's essential for figuring out scenarios like those in our problem, where we want to trace the locus of a point dividing a segment on a parabola.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, utilizes algebra to study geometric problems.
It allows the representation and analysis of geometric figures through algebraic equations.

• This field links geometric shapes and algebraic expressions through the coordinate plane, using formulas and methodologies to solve problems related to positions and distances.
• Coordinate geometry is indispensable when deriving the equation of a locus as it helps transition from a geometric configuration to a numerical solution.

This branch of mathematics is key for understanding and solving descriptions of points, lines, and curves, including parabola, correlating spatial understanding with algebraic operation.