Problem 39 - A Complete Resource Book in Mathematics for JEE Main

Question

A variable plane passes through a fixed point \((a, b, c)\) and meets the coordinate axes in \(A, B, C\). The locus of the point common to the planes through \(A, B, C\) parallel to coordinate planes is (A) \(a y z+b z x+c x y=x y z\) (B) \(a y z+b z x+c x y=2 x y z\) (C) \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (C) \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1\).

Step 1: Equation of the Plane

Let's first write the equation of any plane passing through the point \((a, b, c)\). The general equation of a plane is given by \(x/a + y/b + z/c = 1\).

Step 2: Points on Coordinate Axes

To find the points where the plane intersects the coordinate axes, we set two variables to zero at a time in the plane equation. If \(x\) is set to 0, it intersects y-axis at \(B(0, b, 0)\). Similarly, it intersects z-axis at \(C(0, 0, c)\) and x-axis at \(A(a, 0, 0)\).

Step 3: Equation of Planes Parallel to Axes through A, B, and C

The planes that are parallel to the coordinate planes and pass through A, B, and C can be described as follows. Through point A, a plane parallel to the yz-plane would be \(x = a\). Through point B, parallel to the xz-plane would be \(y = b\), and through point C, parallel to the xy-plane would be \(z = c\).

Step 4: Locus of Common Point

The point \((x, y, z)\) common to these three planes must satisfy all the plane equations: \(x = a\), \(y = b\), and \(z = c\). However, substituting these values leads us to a constraint \(\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = \frac{x}{x} + \frac{y}{y} + \frac{z}{z} = 1\). This relationship governs the locus condition.

Key Concepts

Coordinate Geometry

Coordinate geometry is a fascinating branch of mathematics that blends algebra and geometry to describe the position and relationship of points in space. It uses an algebraic approach to geometrical problems by placing figures in a coordinate plane. This allows mathematicians and students to derive equations that describe geometric shapes, lines, and planes. When we talk about a 'plane,' it refers to a flat, two-dimensional surface extending infinitely. Using coordinates, we can specify a plane using numbers that describe its position in 3D space. In this exercise, coordinate geometry helps us understand where the plane intersects the axes and how those intersection points relate to each other.

Equation of Plane

The equation of a plane is essential for representing a plane in 3D space using mathematical language. In general terms, the equation of a plane can be expressed as \(Ax + By + Cz = D\) or in intercept form as \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\). This intercept form is especially useful for understanding where a plane intersects each axis. In this form, \(a, b,\) and \(c\) refer to the points where the plane intersects each respective axis. This equation can help visualize the plane in 3D because it constrains the relationship between x, y, and z. It's a powerful tool for solving problems that require finding intersection points, distances, and relationships between different planes and lines.

Intercept Form

The intercept form is a specific representation of a plane's equation that clearly shows where a plane intersects with the x, y, and z axes. By writing the equation in the form \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\), it becomes evident that the plane intersects the axes at \(x = a\), \(y = b\), and \(z = c\). This makes it simpler to solve problems involving the intersection of a plane with the coordinate axes, as you can directly see the intercepts. In the context of our exercise, the intercept form is used to identify points A, B, and C, which are the intersection points of the given plane with the x, y, and z axes, respectively. It streamlines calculations and provides a clear geometric interpretation.

Locus Condition

A locus is a collection of points that satisfy certain conditions or rules. The locus condition is all about finding the set of points that meet these specified criteria. In coordinate geometry, a locus is often represented by an equation that the points adhere to. For this exercise, the locus of the point common to the planes through A, B, and C involves finding an equation that all such points \((x, y, z)\) must satisfy. Through understanding the concept of planes parallel to coordinate planes, we deduce that the point common to these planes satisfies the condition \(\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1\). This equation tells us the relationship required for a point to be at the locus of the defined conditions. Such algebraic constrains are crucial for solving complex spatial problems by translating them into manageable equations.