Problem 21 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant \(\lambda\). It passes through a fixed point, which has coordinates \(\begin{array}{ll}\text { (A) }(\lambda, \lambda, \lambda) & \text { (B) }\left(\frac{1}{\lambda}, \frac{1}{\lambda}, \frac{1}{\lambda}\right)\end{array}\) (C) \((-\lambda,-\lambda,-\lambda)\) (D) \(\left(-\frac{1}{\lambda},-\frac{1}{\lambda},-\frac{1}{\lambda}\right)\)

Step-by-Step Solution

Answer

The plane passes through the point \( \left(\frac{1}{\lambda}, \frac{1}{\lambda}, \frac{1}{\lambda}\right) \) (Option B).

Step 1: Understand the Problem

A plane makes intercepts on the X, Y, and Z axes. The sum of the reciprocals of these intercepts is given as a constant \( \lambda \). The problem requires us to determine through which fixed point the plane always passes.

Step 2: Formulate the Equation

The general equation of a plane with intercepts \( a \), \( b \), and \( c \) on the X, Y, and Z axes, respectively, is \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \). According to the problem, \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \lambda \).

Step 3: Define a Fixed Point

Since \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \) holds for any plane with a fixed intercept sum, we assume it passes through an unknown fixed point. Substitute \( x = x_0 \), \( y = y_0 \), \( z = z_0 \) to validate.

Step 4: Substitute the Intercepts Values

Substitute the fixed point coordinates \( x = \frac{1}{\lambda}, y = \frac{1}{\lambda}, z = \frac{1}{\lambda} \) into \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \). Each term adds as \( \lambda \cdot \frac{1}{\lambda} = 1 \). Hence, this point satisfies the equation.

Step 5: Confirm the Fixed Point (Option)

Verify if any other options can give such simplifications. None other than Option B (\(\left(\frac{1}{\lambda}, \frac{1}{\lambda}, \frac{1}{\lambda}\right)\)) satisfies the equation for the given condition consistently.

Key Concepts

Variable Plane

In coordinate geometry, a variable plane is not fixed in space; it can change its position while maintaining a specific property or set of conditions. When dealing with such a plane, especially in relation to its intercepts on coordinate axes, we start by understanding its general representation. A plane can generally be expressed with the equation \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \), where \(a\), \(b\), and \(c\) are the intercepts made by the plane on the X, Y, and Z axes, respectively. This equation tells us how the plane interacts with each axis, and by varying \(a\), \(b\), and \(c\), the plane's position changes too. What's unique about a variable plane is that even though its intercepts can change, it adheres to a specific condition, such as the sum of the reciprocals of the intercepts being a constant.

Intercepts

Intercepts are the points at which a plane or line crosses the axes of a coordinate system. For a plane defined by \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \), \(a\), \(b\), and \(c\) represent the X, Y, and Z intercepts, respectively. These intercepts give insight into how the plane is oriented in the 3D space. When we talk about intercepts in algebra and geometry, we typically want to know where a line or plane meets the axes because these points help us sketch and understand the object’s spatial configuration. In this context, while the intercepts may shift as the plane moves, the condition that their reciprocal sum equals a constant \( \lambda \) provides a specific geometric constraint.

Fixed Point

A fixed point in the realm of coordinate geometry is a stationary point through which a given geometric entity, such as a plane, always passes regardless of how other variables change. In the problem at hand, while the position of the variable plane moves by altering its intercepts, it consistently passes through the point \( \left( \frac{1}{\lambda}, \frac{1}{\lambda}, \frac{1}{\lambda} \right) \). This is confirmed by substituting these coordinates into the plane's intercept equation, showing that they satisfy the equation irrespective of the specific intercepts \(a\), \(b\), and \(c\) as long as their reciprocal condition is met. Hence, this fixed point serves as an anchor or reference that remains unchanged even as the plane shifts.

Reciprocal Sum

The reciprocal sum is the key condition that underlies the relationship between the intercepts of the plane. Mathematically, the reciprocal of a number is \( 1 \) divided by that number. Thus, in our plane equation, the reciprocals are expressed as \( \frac{1}{a}, \frac{1}{b}, \text{and} \frac{1}{c} \). The requirement that these reciprocals sum to a constant \( \lambda \) means:

• \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \lambda \)

By maintaining this equation, even as the intercepts \(a\), \(b\), and \(c\) might individually vary, the overall behavior of the plane remains consistent. This constant reciprocal sum constraint ensures a particular set of geometric properties regardless of the shifts in the plane's position.