Problem 88 - A Complete Resource Book in Mathematics for JEE Main

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: Define the Harmonic Progression Condition

Given that the numbers \(a, b, c\) are in Harmonic Progression (H.P.), it means their reciprocals are in Arithmetic Progression (A.P.). Thus, \(\frac{1}{a}\), \(\frac{1}{b}\), \(\frac{1}{c}\) are in A.P. This implies \( \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \).

Step 2: Use the Harmonic Progression Condition

Rewrite the line equation \( \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \). Substitute for \( \frac{1}{c} \) from the H.P. condition as \( \frac{1}{c} = \frac{2}{b} - \frac{1}{a} \). Substitute in the line equation to get \( \frac{x}{a} + \frac{y}{b} + \frac{2}{b} - \frac{1}{a} = 0 \).

Step 3: Simplify the Equation

Combine the terms in the equation together: \( \frac{x - 1}{a} + \frac{y + 2}{b} = 0 \). This equation describes a family of lines.

Step 4: Find the Fixed Point

A line that passes through a fixed point does so regardless of \(a\) and \(b\) values. Thus, let both coefficients of the terms with \(\frac{1}{a}\) and \(\frac{1}{b}\) cancel each other out independently: \( x - 1 = 0 \) and \( y + 2 = 0 \) which gives us the fixed point \((1, -2)\).

Key Concepts

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference". Here is a simple way to understand it:

• If the sequence starts with the number \(a\), then the next term is \(a + d\), where \(d\) is the common difference.
• The sequence continues as \(a, a + d, a + 2d, \ldots\).

For instance, in the sequence 2, 4, 6, 8, ..., the common difference is 2. When solving problems related to Harmonic Progression, it is helpful to understand Arithmetic Progression because the reciprocals of the numbers in a Harmonic Progression form an Arithmetic Progression. This means that if you have numbers \(a, b, c\) in H.P., the sequence \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) will be in A.P., making the relationship useful for simplifying equations or finding connections.

Fixed Point

In geometry, a fixed point of a function or transformation is a point that remains unchanged when the function is applied. In context of lines, a fixed point is a specific point that lies on every line within a given family of lines.When dealing with an equation like \( \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \), it describes a family of lines dependent on \(a, b, c\). The sequence condition and equation manipulation, as in the original solution, help identify this invariant point. For instance:

• The substitution and simplification steps lead to finding when the equation holds independently of the specific parameters \(a\) and \(b\).
• Setting the line conditions \(x - 1 = 0\) and \(y + 2 = 0\) results in the point \((1, -2)\), which is fixed for the entire family of lines defined by the given equation.

When solutions require identifying a fixed point, it often involves manipulating the equation to be independent of certain variables, checking alignment of coefficients, or directly solving resulting equations.

Equation of a Line

The equation of a line is a critical concept in algebra and geometry, describing a wide range of linear relations between variables. The standard form is typically \( Ax + By + C = 0 \), where \(A, B,\) and \(C\) are constants, describing a straight line with slope and intercept properties.In our specific problem, the equation \( \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \) was used to explore harmonic progression and identify the fixed point for a family of lines. Here are key points about solving line equations:

• The coefficients yourself determine the line's slope and position.
• In problems involving multiple variables, substitution and manipulation changes the line's form to highlight underlying patterns or fixed components.
• Simplifying complex terms can reveal intersections or points common to multiple lines, known as fixed points.

Understanding the arrangement of line equations assists in visualizing relationships on a plane, predicting intersections, and following algebraic transformations that arise in various mathematical contexts.