Problem 103 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(O\) be the vertex and \(Q\) be any point on the parabola, \(x^{2}=8 y\). If the point \(P\) divides the line segment \(O Q\) internally in the ratio \(1: 3\), then the locus of \(P\) is: (A) \(y^{2}=x\) (B) \(y^{2}=2 x\) (C) \(x^{2}=2 y\) (D) \(x^{2}=y\)

Step-by-Step Solution

Answer

The locus of P is \(y^2 = 2x\) (Option B).

Step 1: Understand the Parabola Equation

The given parabola is described by the equation \(x^2 = 8y\). This parabola opens upwards with its vertex at the origin \((0, 0)\). Any point \(Q\) on this parabola can be represented by coordinates \((x, y)\) that satisfy this equation.

Step 2: Find Coordinates of Point Q

Let \(Q = (x_1, y_1)\) be a point on the parabola. Since \(Q\) lies on the parabola \(x^2 = 8y\), it follows that \(y_1 = \frac{x_1^2}{8}\). Therefore, point \(Q\) has coordinates \((x_1, \frac{x_1^2}{8})\).

Step 3: Apply Section Formula to Find P

The point \(P\) divides the segment \(OQ\) in a ratio of \(1:3\). The section formula gives coordinates of \(P\) as: \(P = \left(\frac{1 \times x_1 + 3 \times 0}{1 + 3}, \frac{1 \times \frac{x_1^2}{8} + 3 \times 0}{1 + 3}\right)\). Simplifying, \(P = \left(\frac{x_1}{4}, \frac{x_1^2}{32}\right)\).

Step 4: Establish Locus Equation of P

The coordinates of \(P\) are \(\left(\frac{x_1}{4}, \frac{x_1^2}{32}\right)\). Let \(X = \frac{x_1}{4}\) and \(Y = \frac{x_1^2}{32}\). Relating these:- Since \(Y = \frac{X^2}{8}\), substitute \(x_1 = 4X\) into \(Y = \frac{(4X)^2}{32} = \frac{16X^2}{32} = \frac{X^2}{2}\).- Therefore, the equation for the locus of \(P\) is \(Y^2 = 2X\).

Step 5: Confirm Correct Option

The locus of \(P\) is \(Y^2 = 2X\), which matches option (B). Thus, the correct answer is \(y^2 = 2x\).

Key Concepts

Parabola Equation

A parabola is a symmetrical plane curve which is defined by a specific algebraic equation. In this context, the equation of our parabola is given by \(x^2 = 8y\). This tells us that the parabola opens upwards, with its vertex at the origin \((0, 0)\).

The symmetry of our equation means that for every point \(Q\) on the parabola, the coordinates \((x, y)\) satisfy the equation \(x^2 = 8y\). Therefore, if you know the value of \(x\), you can always find \(y\) using the equation \(y = \frac{x^2}{8}\).

Key characteristics of a parabola

• The vertex, the "turning" point of the parabola, is located at \((0, 0)\) for this particular equation.
• The axis of symmetry is the \(y\)-axis, which means it's vertically aligned.
• It opens upwards because the coefficient of \(y\) is positive in the equation.

Understanding these features is crucial for solving more complex questions involving parabolas, like finding other points on or characteristics of the parabola.

Section Formula

The section formula is a powerful tool in coordinate geometry. It helps determine the coordinates of a point that divides a given line segment joining two points in a specified ratio. For a line segment \(OQ\) with endpoints \(O(x_0, y_0)\) and \(Q(x_1, y_1)\), the formula is used when the point \(P(x, y)\) divides \(OQ\) internally in the ratio \(m:n\).

Using the section formula, the coordinates of \(P\) are given by:\[x = \frac{mx_1 + nx_0}{m+n}, \quad y = \frac{my_1 + ny_0}{m+n}\]In our specific problem, \(O\) is at the origin \((0, 0)\) and \(Q\) is \((x_1, y_1)\) on the parabola. \(P\) divides the segment \(OQ\) in the ratio \(1:3\). So, applying the section formula:

• For \(x\)-coordinate: \(x = \frac{1 \cdot x_1 + 3 \cdot 0}{4} = \frac{x_1}{4}\)
• For \(y\)-coordinate: \(y = \frac{1 \cdot y_1 + 3 \cdot 0}{4} = \frac{y_1}{4}\)

This formula provides a systematic way to find a point within a segment, essential for determining locus points.

Dividing Line Segment

Dividing a line segment involves finding a new point \(P\) that lies somewhere along the line segment between two endpoints \(O\) and \(Q\). To determine where exactly \(P\) is, you'd use the aforementioned section formula. Dividing internally means that \(P\) lies between \(O\) and \(Q\).

When \(P\) divides \(OQ\) in the ratio of \(1:3\), it means the part of the line segment from \(O\) to \(P\) is one-quarter of the entire segment. Consequently, \(P\)'s coordinates can help us establish a relationship or a path, known as the locus.

• With \(O = (0, 0)\), \(x_0 = 0\) and \(y_0 = 0\).
• For \(Q = (x_1, y_1)\), substitute \(y_1 = \frac{x_1^2}{8}\).

Thus, the coordinates for \(P\) become \((\frac{x_1}{4}, \frac{x_1^2}{32})\). After finding \(P\)'s coordinates, further algebraic manipulation helps us find that the locus of \(P\) adheres to the equation \(Y^2 = 2X\). This observation stems from substituting and simplifying the expressions for the coordinates.