Problem 41 - A Complete Resource Book in Mathematics for JEE Main

Question

If the median through \(A\) of a \(\Delta A B C\) having vertices \(A\) \(\equiv(2,3,5), B \equiv(-1,3,2)\) and \(C \equiv(\lambda, 5, \mu)\) is equally inclined to the axes, then (A) \(\lambda=7\) (B) \(\mu=10\) (C) \(\lambda=10\) (D) \(\mu=7\)

Step-by-Step Solution

Answer

(A) \(\lambda=7\) and (B) \(\mu=10\).

Step 1: Find the Midpoint of BC

The midpoint of the line segment connecting points \(B(-1, 3, 2)\) and \(C(\lambda, 5, \mu)\) can be calculated using the formula for the midpoint: \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)\]Substitute the values for \(B\) and \(C\):\[M = \left(\frac{-1 + \lambda}{2}, \frac{3 + 5}{2}, \frac{2 + \mu}{2}\right) = \left(\frac{-1 + \lambda}{2}, 4, \frac{2 + \mu}{2}\right)\]

Step 2: Determine the Direction Ratio of Median AM

The direction ratios of the line segment (median) \(AM\) from point \(A(2, 3, 5)\) to the midpoint \(M\left(\frac{-1 + \lambda}{2}, 4, \frac{2 + \mu}{2}\right)\) are given by:\[\left( \frac{-1 + \lambda}{2} - 2,\ 4 - 3,\ \frac{2 + \mu}{2} - 5 \right)\]This simplifies to:\[\left( \frac{-5 + \lambda}{2},\ 1,\ \frac{-8 + \mu}{2} \right)\]

Step 3: Application of Equal Inclination Condition

The condition that a line is equally inclined to all axes means that its direction ratios are proportional. Hence, we equate the terms: \[\frac{-5 + \lambda}{2} = 1 = \frac{-8 + \mu}{2}\]

Step 4: Solve for \(\lambda\) and \(\mu\)

From \(\frac{-5 + \lambda}{2} = 1\), simplify to find \(\lambda\):\[-5 + \lambda = 2 \implies \lambda = 7\]From \(\frac{-8 + \mu}{2} = 1\), simplify to find \(\mu\):\[-8 + \mu = 2 \implies \mu = 10\]

Step 5: Validate the Solution

Plug \(\lambda = 7\) and \(\mu = 10\) back into the direction ratios to confirm they maintain consistency:\[\left( \frac{-5 + 7}{2},\ 1,\ \frac{-8 + 10}{2} \right) = (1, 1, 1)\]All direction ratios are equal, validating that our solution confirms the line being equally inclined to all axes.

Key Concepts

Direction Ratios

Direction ratios are crucial in understanding the orientation of a line in a three-dimensional space. These values give us insight into how steep or angled a line is concerning each axis. In simpler words, direction ratios can be seen as the vector components of a line segment.
For example, when we calculated the direction ratios of the median line from point A to the midpoint M, it helped us understand how this line inclines or aligns along the x, y, and z axes.
To determine the direction ratios for any line segment, we simply take the differences of corresponding coordinates. This process allowed us to find \(( \frac{-5 + \lambda}{2}, 1, \frac{-8 + \mu}{2} )\). Understanding these values can tell you a lot about the line’s orientation in space.

Midpoint Formula

The midpoint formula is a fundamental tool in geometry that helps us find the center point of a line segment. It not only gives spatial understanding but also helps in solving problems involving symmetry or balance.
The formula is straightforward: for any two points \( (x_1, y_1, z_1)\) and \( (x_2, y_2, z_2)\), the midpoint M is given by:

• \(( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} )\)

By using this formula, we calculated the midpoint \(M\) of segment \(BC\) as \( \left( \frac{-1 + \lambda}{2}, 4, \frac{2 + \mu}{2} \right)\) which served as a crucial step in deriving the direction ratios.
Finding midpoints is essential, especially for determining other properties like the equation of medians or bisectors in triangles.

Inclination of a Line

The inclination of a line in geometry refers to its angle respective to a particular axis. Generally, many problems in solid geometry involve calculating how a line leans or tilts in space.
For a line to be equally inclined to the coordinate axes, its direction ratios should be in proportion. This is key to understanding the unique alignment of a line where each direction is balanced or equivalent in its lean.
In our exercise, the condition of equal inclination confirmed the direction ratios being equal or proportional, i.e., \( \frac{-5 + \lambda}{2} = 1 = \frac{-8 + \mu}{2} \). Recognizing this inclination condition helps ensuring certain geometric properties such as symmetry or parallelism with bisected planes.

Equally Inclined

When a line is equally inclined to all axes, it indicates a specific and uniform orientation. Such a line makes equal angles with all the Cartesian coordinate lines.
This concept of equally inclined lines was applied in our median calculation where solving for \(\lambda\) and \(\mu\) yielded direction ratios all equaling \(1\).

• This condition brought us to find solutions \(\lambda = 7\) and \(\mu = 10\), ensuring that all ratios balanced perfectly.

This kind of spatial orientation is significant when analyzing geometric shapes because it reveals inherent symmetry and balance, contributing to understanding the structure and distribution of points in space.