Problem 120 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The distance of the point \((1,-5,9)\) from the plane \(x-y+\) \(z=5\) measured along a straight \(x=y=z\) is \(\quad\) [2011RS] (a) \(10 \sqrt{3}\) (b) \(5 \sqrt{3}\) (c) \(3 \sqrt{10}\) (d) \(3 \sqrt{5}\)

Step-by-Step Solution

Answer

The distance is \(10 \sqrt{3}\).

Step 1: Understand the Given Problem

We need to calculate the distance from the point \((1, -5, 9)\) to the plane \(x - y + z = 5\), with the condition that this distance is measured along a line where \(x = y = z\).

Step 2: Find the Direction Vector

Since the line has the condition \(x = y = z\), the direction vector of the line can be represented as \(\mathbf{d} = (1, 1, 1)\). This means any movement along the line changes each coordinate equally.

Step 3: Parametrize the Line

We can represent a point \((x, y, z)\) on the line as \((1 + t, -5 + t, 9 + t)\), where \(t\) is the parameter. This equation shows it's a line through the point \((1, -5, 9)\) in the direction of \(\mathbf{d}\).

Step 4: Find Intersection with the Plane

Substitute the parametric line equations into the plane equation. Plug \(x = 1 + t\), \(y = -5 + t\), \(z = 9 + t\) into \(x - y + z = 5\), we get:\[(1 + t) - (-5 + t) + (9 + t) = 5.\]

Step 5: Simplify the Equation

Simplify the equation:\[1 + t + 5 - t + 9 + t = 5.\]\[15 + t = 5.\]Solve for \(t\):\[t = 5 - 15 = -10.\]

Step 6: Substitute Back to Find Coordinates

Substitute \(t = -10\) back into the parametric equations to find the coordinates where the line intersects with the plane: \(x = 1 - 10 = -9\),\(y = -5 - 10 = -15\),\(z = 9 - 10 = -1\). The point of intersection is \((-9, -15, -1)\).

Step 7: Calculate the Distance

The distance \(D\) from \((1, -5, 9)\) to \((-9, -15, -1)\) can be calculated using the distance formula:\[D = \sqrt{(-9 - 1)^2 + (-15 + 5)^2 + (-1 - 9)^2}.\]Simplifying:\[D = \sqrt{(-10)^2 + (-10)^2 + (-10)^2} = \sqrt{300} = 10\sqrt{3}.\]

Step 8: Confirm with Given Choices

Compare the calculated distance \(10\sqrt{3}\) with the provided options. The correct answer is option (a) \(10\sqrt{3}\).

Key Concepts

Distance from a Point to a Plane

In coordinate geometry, the distance from a point to a plane is a fundamental concept. This distance helps us measure how far a point is from a given plane in three-dimensional space. To calculate this distance, we generally draw a perpendicular from the point to the plane.

However, in some cases, like our original problem, the distance is calculated along a specific direction rather than directly perpendicular. When the line is defined by a parameter where each coordinate changes equally, such as where \(x = y = z\), the direction is different from the perpendicular norm usually used. Here, we had to find the point on the plane where the line from the given point intersected it, and then calculate the distance along this specified direction.

Parametric Equations

Parametric equations are a way of expressing the coordinates of the points that make up a curve or a surface. These equations use one or more parameters to represent the coordinates along a line.

• Setup: For a line, we express the coordinates \((x, y, z)\) using a parameter, typically denoted as \(t\). In our problem, this involved the equations \(x = 1 + t\), \(y = -5 + t\), and \(z = 9 + t\).
• Usage: These equations are useful for finding points along the line and are particularly handy when determining where a line intersects with a plane.
  Consider the parameter \(t\) as a slider; moving it changes the point along the line, showcasing locations wherever \(t\) is considered.

They are a critical mathematical tool for precisely describing spatial relationships and movements.

Direction Vectors

In analyzing lines in space, direction vectors are vital for understanding the orientation and behavior of the line.

• Definition: A direction vector, \(\mathbf{d}\), is a vector that represents the direction of a line. In our scenario, the condition \(x = y = z\) leads to the direction vector \(\mathbf{d} = (1, 1, 1)\). It signifies that any unit movement results in simultaneous and equal changes in \(x\), \(y\), and \(z\).
• Function: This vector is crucial for defining how one can "move" along the line, which is fundamental when determining movement towards planes or other geometrical figures.
  Direction vectors guide calculations and intersections in vector mathematics, showing how lines travel through space.

Remember, direction vectors lay the groundwork for many calculations involving vector magnitude and direction, allowing us to explore more complex geometrical problems.