Problem 42 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If non zero numbers \(a, b, c\) are in H.P., then the straight line \(\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0\) always passes through a fixed point. That point is [2005] (a) \((-1,2)\) (b) \((-1,-2)\) (c) \((1,-2)\) (d) \(\left(1,-\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The fixed point is \((-1, 2)\).

Step 1: Understanding Harmonic Progression (H.P.)

In a Harmonic Progression (H.P.), the reciprocals of the terms are in Arithmetic Progression (A.P.). For numbers \(a, b, c\) in H.P., this means their reciprocals are in A.P., i.e., \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) form an A.P. We can express this as \( \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \).

Step 2: Rearranging the Line Equation

The given line equation is \( \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \). We want to find if this line passes through a fixed point. Substitute the equation with the condition from Step 1. Simplify to find a constant term that implies a fixed point.

Step 3: Apply H.P. Condition to the Line Equation

Using the H.P. condition from Step 1, substitute \( \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \) into the line equation. Rewriting gives \( \frac{x}{a} + \frac{y}{b} + \left(\frac{2}{b} - \frac{1}{a}\right) = 0 \). Simplify this to find expressions for \(x\) and \(y\).

Step 4: Solve for the Line Passing through a Fixed Point

Simplifying, \( \frac{x}{a} + \frac{y - 2}{b} = 0 \), implies a parallel condition. For a fixed intersection point, let this point hold for general \(x\) and \(y\). Solving this for particular fixed coordinates yields a specific solution.

Step 5: Identifying the Fixed Point

By solving the adjusted equation \( \frac{x}{a} + \frac{y - 2}{b} = 0 \), we find the coordinate values where the line always holds. The consistent solution across the rearrangements reveals the fixed intersection point to be \((-1, 2)\). Hence, the line always passes through this point.

Key Concepts

Fixed Point

In geometry, a fixed point is a point through which a geometric figure such as a line always passes, no matter the adjustments applied due to other conditions or constraints. In this context, examining a fixed point implies finding a specific location in the coordinate plane consistent across changes in the variables or parameters involved.
For our line equation, \( \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \), the task is to identify a point through which this line consistently passes, regardless of the values of \( a, b, \text{and} \ c \) provided they satisfy the condition of Harmonic Progression. Step by step, we derive a fixed point, essentially anchoring the geometric figure within a coordinate system.
This fixed point highlights how certain relationships among variables lead to a stable and predictable outcome, visible here as the point \((-1, 2)\). Through algebraic manipulation using the condition of Harmonic Progression, we establish that no matter what values are chosen for \( a, b, \text{and} \ c \) (as long as they remain non-zero and satisfy the H.P. condition), the line will intersect at the fixed point \((-1, 2)\), making it a fundamental part of the solution.

Arithmetic Progression

Arithmetic Progression (A.P.) is a sequence of numbers such that the difference between the consecutive terms is constant. This sequence forms a straightforward pattern, and calculating unknown terms is relatively uncomplicated with its fixed increment between terms.
In mathematics, recognizing such patterns in sequences allows us for precise predictions and simplifications. In the problem at hand, the reciprocals of the terms in Harmonic Progression (H.P.) need to establish an A.P. This involves verifying that \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) yields a consistent difference, pointing to sequences such as \( \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \).
This transforms the complexity of dealing with harmonic sequences into the more straightforward arithmetic progression scenario. It simplifies handling the unknowns and calculating further transformations, insights mirrored in sequences that follow a predictable additive pattern.

Line Equation in Analytical Geometry

Analytical geometry involves using algebraic methods to solve geometric problems. Line equations, central to this field, describe the locations and directions of lines in coordinate planes, allowing us to determine intersections, directionality, and fixed points.
The generic line equation format, like \( Ax + By + C = 0 \), becomes an essential tool for this purpose. In our exercise, the equation is given as \( \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \), expressing a line in relation to multiple variables. Analytical manipulation in the context of understanding harmonic and arithmetic progressions helps deduce the line's path.
This line's intersection across all valid values of the coefficients arises from deriving consistent geometric properties through algebra. The analytic approach provides not only solutions but also insights into the inherent symmetry and balance within these mathematical constructs.