Problem 32 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If two lines \(L_{1}\) and \(L_{2}\) in space, are defined by \(L_{1}=\\{x=\sqrt{\lambda} y+(\sqrt{\lambda}-1), z=(\sqrt{\lambda}-1) y+\sqrt{\lambda}\\}\) and \(L_{2}=\\{x=\sqrt{\mu} y+(1-\sqrt{\mu}), z=(1-\sqrt{\mu}) y+\sqrt{\mu}\\}\) then \(L_{1}\) is perpendicular to \(L_{2}\), for all non-negative reals \(\lambda\) and \(\mu\), such that : \(\quad\) [Online April 23, 2013] (a) \(\sqrt{\lambda}+\sqrt{\mu}=1\) (b) \(\lambda \neq \mu\) (c) \(\lambda+\mu=0\) (d) \(\lambda=\mu\)

Step-by-Step Solution

Answer

(a) \(\sqrt{\lambda} + \sqrt{\mu} = 1\).

Step 1: Identify the Direction Vectors

Firstly, understand the geometric meaning of the lines. Lines in three-dimensional space can be expressed in vector form. The direction vector of line \(L_1\) is given by the coefficients of \(y\) in the expressions of \(x\) and \(z\), thus the direction vector is \((\sqrt{\lambda}, \sqrt{\lambda} - 1)\). Similarly, the direction vector for line \(L_2\) is \((\sqrt{\mu}, 1 - \sqrt{\mu})\).

Step 2: Determine the Condition for Perpendicularity

Two lines are perpendicular if their direction vectors are orthogonal. This is achieved when their dot product equals zero. Compute the dot product of the direction vectors: \((\sqrt{\lambda}, \sqrt{\lambda} - 1) \cdot (\sqrt{\mu}, 1 - \sqrt{\mu})\).

Step 3: Calculate the Dot Product

Calculate the dot product: \(\sqrt{\lambda} \cdot \sqrt{\mu} + (\sqrt{\lambda} - 1) \cdot (1 - \sqrt{\mu}) = \sqrt{\lambda} \sqrt{\mu} + \sqrt{\lambda} - \sqrt{\lambda} \sqrt{\mu} - 1 + \sqrt{\mu}\). Simplify this to \(\sqrt{\lambda} - 1 + \sqrt{\mu}\).

Step 4: Set Dot Product to Zero

For the lines to be perpendicular, set the simplified expression to zero: \(\sqrt{\lambda} - 1 + \sqrt{\mu} = 0\). Rearrange to find \(\sqrt{\lambda} + \sqrt{\mu} = 1\).

Step 5: Conclusion

Examine the provided options and identify which matches the derived condition. The condition \(\sqrt{\lambda} + \sqrt{\mu} = 1\) corresponds to option (a). Therefore, lines \(L_1\) and \(L_2\) are perpendicular under this condition.

Key Concepts

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). In vector geometry, two lines can be represented by direction vectors. When dealing with lines in three-dimensional space, these vectors help us define the orientation of each line. To determine if two lines are perpendicular, you can use these direction vectors. Two lines are perpendicular if their direction vectors are orthogonal. This means that the angle between the two vectors is 90 degrees, or mathematically, that their dot product is equal to zero. In our example, lines \(L_1\) and \(L_2\) have specific conditions under which they are perpendicular, which involves the values of \(\lambda\) and \(\mu\), ensuring their dot product sums to zero.

Direction Vectors

Direction vectors are crucial for understanding the alignment of lines in vector geometry. Typically expressed in the form \((a, b, c)\) in three-dimensional space, they indicate the direction a line travels. From the given problem, each line \(L_1\) and \(L_2\) has a specific direction vector derived from the coefficients of \(y\) present in their equations. For line \(L_1\), the direction vector is \((\sqrt{\lambda}, \sqrt{\lambda} - 1)\), while for line \(L_2\), it is \((\sqrt{\mu}, 1 - \sqrt{\mu})\). These vectors essentially show the directional movement of each line in two or three dimensions. Determining the perpendicularity of the lines involves analyzing these direction vectors and their relationship through the dot product.

Dot Product

The dot product is a key operation in vector mathematics used to determine the orthogonality of vectors. It involves multiplying corresponding components of two vectors and summing the results, giving a scalar output. For example, for vectors \((a, b)\) and \((c, d)\), the dot product is \(a \cdot c + b \cdot d\).

In our problem, the dot product of the direction vectors \((\sqrt{\lambda}, \sqrt{\lambda} - 1)\) and \((\sqrt{\mu}, 1 - \sqrt{\mu})\) is \(\sqrt{\lambda} \cdot \sqrt{\mu} + (\sqrt{\lambda} - 1) \cdot (1 - \sqrt{\mu})\). Simplifying this expression leads to \(\sqrt{\lambda} - 1 + \sqrt{\mu}\). For the lines to be perpendicular, this must equal zero. This condition gives us the relationship \(\sqrt{\lambda} + \sqrt{\mu} = 1\), solving the problem condition for perpendicularity.