The two lines x=a y+b, z=c y+d ; and x=a^' y+b^', z =c^' y+d^' are perpendicular to each other i

Name: Problem 55, Chapter 22: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-10-21T05:51:33-08:00
Duration: 2 min 4 s
Channel: Varsha Aggarwal
Description: The two lines x=a y+b, z=c y+d ; and x=a^' y+b^', z =c^' y+d^' are perpendicular to each other if [2 0 0 6] (A) a a^'+c c^'=-1 (B) a a^'+c c^'=1 (C) (a)/(a^')+(c)/(c^')=-1

step 1 So these are equation of line. So from first, from first y equal x minus b upon a and from z equ

Question

The two lines $x=a y+b, z=c y+d ;$ and $x=a^{\prime} y+b^{\prime}, z$ $=c^{\prime} y+d^{\prime}$ are perpendicular to each other if $[\mathbf{2 0 0 6}]$ (A) $a a^{\prime}+c c^{\prime}=-1$ (B) $a a^{\prime}+c c^{\prime}=1$ (C) $\frac{a}{a^{\prime}}+\frac{c}{c^{\prime}}=-1$

   The two lines $x=a y+b, z=c y+d ;$ and $x=a^{\prime} y+b^{\prime}, z$ $=c^{\prime} y+d^{\prime}$ are perpendicular to each other if $[\mathbf{2 0 0 6}]$
(A) $a a^{\prime}+c c^{\prime}=-1$
(B) $a a^{\prime}+c c^{\prime}=1$
(C) $\frac{a}{a^{\prime}}+\frac{c}{c^{\prime}}=-1$

A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar 2019 Edition

Chapter 22, Problem 55

Instant Answer

Solved by Expert Varsha Aggarwal

10/21/2021

Step 1

We can rewrite these lines in the standard form as follows: \[ \frac{x-b}{a} = y = \frac{z-d}{c} \quad \text{and} \quad \frac{x-b'}{a'} = y = \frac{z-d'}{c'} \] Show more…

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The two lines $x=a y+b, z=c y+d ;$ and $x=a^{\prime} y+b^{\prime}, z$ $=c^{\prime} y+d^{\prime}$ are perpendicular to each other if $[\mathbf{2 0 0 6}]$ (A) $a a^{\prime}+c c^{\prime}=-1$ (B) $a a^{\prime}+c c^{\prime}=1$ (C) $\frac{a}{a^{\prime}}+\frac{c}{c^{\prime}}=-1$

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Key Concepts

Parametric Representation of Lines

This concept involves expressing a line using a parameter, typically resulting in equations for each coordinate in terms of that parameter. It provides a straightforward way to identify the direction vector of the line by examining the coefficients of the parameter.

Direction Vectors

A direction vector indicates the orientation of a line in space and is derived from the coefficients of the free parameter in the line's parametric equations. It is crucial in comparing the orientations of different lines, especially when analyzing their parallelism or orthogonality.

Dot Product and Orthogonality

In Euclidean geometry, the dot product of two vectors being zero is the criterion for the vectors to be perpendicular to each other. This concept is widely used to determine the perpendicularity of lines by calculating the dot product of their direction vectors and setting it equal to zero.

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