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Chapter 4: Problem 27

A variable plane passes through a fixed point \((a, b, c)\) and meets the coordinate axes in \(A\), B, C. Locus of the point common to the planes through \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and parallel to coordinate planes, is (1) \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1\) (2) \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) (3) \(a x+b y+c z=1\) (4) \(a x+b y+c z=2\) (5) \(\frac{\mathrm{a}}{\mathrm{x}}+\frac{\mathrm{b}}{\mathrm{y}}+\frac{\mathrm{c}}{\mathrm{z}}=2\)

Short Answer

Expert verified
The correct option is \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1 \).

Step by step solution

01

Understand the problem

A variable plane passes through the fixed point \( (a, b, c) \) and intersects the coordinate axes at points \((A, B, C)\). The goal is to find the locus of the point common to the planes passing through \(A, B, C\) and parallel to the coordinate planes.
02

Equation of the plane

Assume the equation of the plane as \( \frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1 \). This plane intersects the x-axis at \(A (p, 0, 0)\), the y-axis at \(B (0, q, 0)\), and the z-axis at \(C (0, 0, r)\).
03

Plane passing through \((a, b, c)\)

Since the plane passes through the fixed point \( (a, b, c) \), substitute these values into the plane equation: \(\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 1\).
04

Locus of the common point

To find the locus of the point common to the planes through \( (a, b, c) \) and parallel to the coordinate planes, recognize that the point remains constant when planes change independent of coordinate planes.
05

Identifying the correct option

Translate the given condition into an equation: \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1\). This matches option (1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Plane
In coordinate geometry, understanding a 'variable plane' is crucial. A variable plane is a plane that can change its position or orientation. It is different from a fixed plane, which remains constant. In our problem, the plane passes through a fixed point \( (a, b, c) \) but varies by changing its intersection with the coordinate axes. The equation for such a variable plane is often expressed as: \[ \frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1 \]. Here, \( p, q, \) and \( r \) are the intersections on the x, y, and z axes, respectively. Such planes help in solving many complex problems in three-dimensional geometry. The key part is that even though the plane changes, the fixed point \( (a, b, c) \) ensures that it retains some consistency.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, uses algebraic equations to describe geometric principles. It helps to locate the position of points in a plane. In our problem, the plane's equation is linked to the coordinate axes intersections. This means the plane meets the x-axis at A(p, 0, 0), the y-axis at B(0, q, 0), and the z-axis at C(0, 0, r). By substituting these points into the plane formula and ensuring it passes through the fixed point (a, b, c), the relation becomes: \[ \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 1 \]. This relation is the foundation to finding the locus, pivotal to coordinate geometry. Additionally, understanding how to manipulate these coordinates and equations is essential for solving more advanced problems.
Intersection with Axes
The 'intersection with axes' concept is fundamental in coordinate geometry. When a plane intersects coordinate axes, it results in specific points on each axis. In our variable plane example, it intersects the x-axis at A(p,0,0), the y-axis at B(0,q,0), and the z-axis at C(0,0,r). These intersections help to define the plane's location in three-dimensional space. Because the variable plane passes through a fixed point (a, b, c), substituting these intersections in the plane equation is crucial. You derive the plane's equation and subsequently the locus of the point common to planes parallel to coordinate planes. Conclusively, recognizing these points of intersections simplifies understanding the behavior and position of the plane.

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