Problem 47 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The line \(x+y=1\) meets \(\mathrm{x}\)-axis at \(A\) and \(\mathrm{y}\)-axis at \(B . P\) is the mid-point of \(A B \cdot P_{1}\) is the foot of the perpendicular from \(P\) to \(O A ; M_{1}\) is that from \(P_{1}\) to \(O P ; P_{2}\) is that from \(M_{1}\) to \(O A\) and so on. If \(P_{n}\) denotes the foot of the \(n\)th perpendicular on \(O A\) from \(M_{n-1}\), then \(O P_{n}\) is equal to (A) \(\frac{1}{2^{n}}\) (B) \(\frac{1}{2^{n-1}}\) (C) \(\frac{1}{2^{n-2}}\) (D) none of these

Step-by-Step Solution

Answer

Answer: (A) \(\frac{1}{2^{n}}\).

Step 1: Determine Points A and B

The line equation given is \(x+y=1\). To find where it intersects the \(x\)-axis, set \(y=0\), giving \(x+0=1\) or \(x=1\). Thus, point \(A\) is \((1,0)\). For the intersection with the \(y\)-axis, set \(x=0\), yielding \(0+y=1\) or \(y=1\). Thus, point \(B\) is \((0,1)\).

Step 2: Find Midpoint P of AB

The midpoint formula \(P = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) can be used to find point \(P\). Substituting \(A(1,0)\) and \(B(0,1)\), we get: \(P = \left(\frac{1+0}{2}, \frac{0+1}{2}\right) = (0.5, 0.5)\).

Step 3: Calculate P1: Foot of Perpendicular from P to OA

Point \(P_1\) is the foot of a perpendicular from \(P(0.5, 0.5)\) to the line segment \(OA\), where \(O = (0,0)\) and \(A = (1,0)\). Since \(OA\) is parallel to the \(x\)-axis, \(P_1\) has the same \(x\)-coordinate as \(P\), i.e., \(P_1 = (0.5, 0)\).

Step 4: Determine M1 and Next Perpendiculars

From \(P_1(0.5,0)\), M1 is the foot of the perpendicular to \(OP\). Since \(OP\) has a slope of 1, the perpendicular slope will be -1. The line through \(P_1(0.5,0)\) with this slope is \(y = -x + 0.5\). Set \(y=0\) for the point on the \(x\)-axis, yielding \(x=0.5\). So \(M_1 = (0.5,0)\).

Step 5: Calculate P2 and Verify Pattern

The same process applies for \(P_2\) from \(M_1\). Now, \(P_2\) lies on \(OA\) and considering the geometric pattern, the \(x\)-coordinate becomes repeatedly halved. Thus \(P_2 = (0.25,0)\).

Step 6: Formulate Function for Pn

The pattern \((1, 0), (0.5, 0), (0.25, 0), ...\) suggests that the \(x\)-coordinate of \(P_n\) is \(x = \frac{1}{2^n}\). So for each \(n\), \(O P_n = \frac{1}{2^n}\).

Key Concepts

Perpendicular Distance

In coordinate geometry, the concept of perpendicular distance is essential to understanding how we measure the shortest path between a point and a line. When determining the foot of the perpendicular from a point to a line, you're essentially finding the point on the line closest to the given point. This closest point forms a perpendicular angle (90 degrees) with the initial point.
In the given problem, for instance, the line segment OA, which lies on the x-axis, receives a foot of perpendicular from point P, found at P1. This essentially means that if you were to draw a line from P to OA, forming a right angle with the line OA, it would touch OA at P1.

• To find such coordinates, you usually adjust one coordinate value (depending on which axis-aligned line you're perpendicular to) to match that of the line, keeping it angularly precise.
• Thus, understanding this concept involves analyzing angles and leveraging coordinate points to manipulate equations and achieve perpendicularity in geometry.

Midpoint Formula

The midpoint formula provides a method to find a point exactly halfway between two other points in a coordinate plane. If you have two points, A (x_1, y_1) and B (x_2, y_2), the midpoint P is given by the formula\[P = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).\]This formula works by taking an average of the x-coordinates and y-coordinates of the given points, acting as a balancing point or a center.
In the context of the problem, the points A and B were determined with ease, using the intercepts on the axes. The midpoint P—calculated using these coordinates—represents the average position between A and B.

This calculation is straightforward as it involves simple arithmetic, ideal for finding symmetry or centers, not only geometrically, in figures but also in problems involving motion or average positions.

Geometric Sequence

A geometric sequence is a number series where each term is derived by multiplying the previous term by a constant factor. This sequence is integral in distinguishing patterns or growth systems and is represented as follows:
Suppose the first term is denoted by \( a \) and the common ratio by \( r \), the nth term in a geometric sequence can be given by the expression:\[a_{n} = a \cdot r^{(n-1)}\]In this exercise, the sequence emerges in determining each subsequent coordinate \( P_n \). Each new position signifies a halving of the x-coordinate from the previous point. So given that the process starts at \((1,0)\), every subsequent foot is at half the distance along the segment line, forming a sequence.

• The geometric series pattern here is represented by continually dividing the x-coordinate by 2, shown in the sequence: \( \frac{1}{2^1} \), \( \frac{1}{2^2} \), \( \frac{1}{2^3} \)... resulting in each \( n^{th} \) position becoming \( \frac{1}{2^n} \).
• The visualization of such patterns and their formulations helps simplify complex geometrical problems into recognizable, easy-to-calculate expressions, making them invaluable in solving coordinate geometry challenges.