Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

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 Midpoint of Side BC

The midpoint, \(M\), of side \(BC\) is given by the formula \( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \). Substituting the coordinates of \(B\) and \(C\), we get \( M = \left( \frac{-1 + \lambda}{2}, \frac{3 + 5}{2}, \frac{2 + \mu}{2} \right) = \left( \frac{-1 + \lambda}{2}, 4, \frac{2 + \mu}{2} \right) \).

Step 2: Find Direction Ratios of the Median AM

The direction ratios of the vector \(AM\) are given by the differences in their coordinates: - \(x\)-component: \(\frac{-1 + \lambda}{2} - 2 = \frac{-5 + \lambda}{2}\) - \(y\)-component: \(4 - 3 = 1\) - \(z\)-component: \(\frac{2 + \mu}{2} - 5 = \frac{-8 + \mu}{2}\)

Step 3: Set Direction Ratios Equal for AM

For the median to be equally inclined, the magnitude of the direction ratios should be the same. Therefore, \(|\frac{-5 + \lambda}{2}| = |1| = |\frac{-8 + \mu}{2}|\). Solve these equations separately to find \(\lambda\) and \(\mu\).

Step 4: Solve \(|\frac{-5 + \lambda}{2}| = 1\) for \(\lambda\)

Solve the equation \(\frac{-5 + \lambda}{2} = 1\) and \(\frac{-5 + \lambda}{2} = -1\):1. \(\frac{-5 + \lambda}{2} = 1 \Rightarrow -5 + \lambda = 2 \Rightarrow \lambda = 7\)2. \(\frac{-5 + \lambda}{2} = -1 \Rightarrow -5 + \lambda = -2 \Rightarrow \lambda = 3\)

Step 5: Solve \(|\frac{-8 + \mu}{2}| = 1\) for \(\mu\)

Solve the equation \(\frac{-8 + \mu}{2} = 1\) and \(\frac{-8 + \mu}{2} = -1\):1. \(\frac{-8 + \mu}{2} = 1 \Rightarrow -8 + \mu = 2 \Rightarrow \mu = 10\)2. \(\frac{-8 + \mu}{2} = -1 \Rightarrow -8 + \mu = -2 \Rightarrow \mu = 6\)

Step 6: Determine Correct Option

The values satisfiying the property that the median is equally inclined to the axes are \(\lambda = 7\) and \(\mu = 10\). These correspond to options (A) \(\lambda=7\) and (B) \(\mu=10\).

Key Concepts

3D Coordinate Geometry

3D Coordinate Geometry extends our usual plane geometry into three dimensions, adding depth to the familiar two-dimensional space. Instead of just 'x' and 'y' coordinates, points in space have three coordinates, \(x, y, z\), to describe their positions.

• **Point Coordinates:** Each point is denoted as \( (x, y, z) \), where **x**, **y**, and **z** represent the position along each axis.
• **Axes:** The three axes, **x**, **y**, and **z**, meet at a single point called the origin, \(O(0, 0, 0)\).
• **Planes:** Three primary planes divide the 3D space: the **xy-plane**, **yz-plane**, and **zx-plane**.

In the exercise provided, the points \(A\), \(B\), and \(C\) are positioned in a 3-dimensional space. You calculated the midpoint of side \(BC\), using the coordinates from \(B\) and \(C\), which is essential for understanding the structure and relationships of a 3D triangle.

Vectors

Vectors are essential in 3D coordinate geometry as they represent both magnitude and direction. A vector in 3D space can be visualized as an arrow pointing from one point to another, showing direction and length.

• **Vector Notation:** A vector from point \(A(x_1, y_1, z_1)\) to \(B(x_2, y_2, z_2)\) is denoted as \(\overrightarrow{AB}\).
• **Components of a Vector:** The vector has components calculated as \((x_2-x_1, y_2-y_1, z_2-z_1)\).

In the problem, you computed the vector \(\overrightarrow{AM}\), representing the median through point \(A\) to the midpoint \(M\) of \(BC\). This vector gives us direction ratios that indicate how much the vector moves in each \axis direction.

Direction Ratios

Direction ratios are numbers that are proportional to the components of a vector and give insight into the direction the vector points. For a vector \((l, m, n)\), the direction ratios are the values \(l\), \(m\), and \(n\).

• **Equal Direction Ratios:** A line is equally inclined to the axes if the absolute values of its direction ratios are the same. This implies symmetry along the three-dimensional space.
• **Calculation:** In our exercise, you worked out the direction ratios for \(\overrightarrow{AM}\) as \(\frac{-5 + \lambda}{2}\), \(1\), and \(\frac{-8 + \mu}{2}\).

By equating these values to ensure they are all equal, you found the constraints \(|\frac{-5 + \lambda}{2}| = 1| = |\frac{-8 + \mu}{2}|\). Solving these helps deduce the correct values for \(\lambda\) and \(\mu\), which guarantee the median is equally inclined to the axes, thus satisfying the problem condition.