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  • I need help with - If the lines and are perpendicular, then: - Three Dimensional Geometry - JEE Main

If the lines x=ay+b, z=cy+z and x={a}'z+{b}',y = {c}'z+{d}' are perpendicular, then:

  • Option 1)

     

    c{c}'+a+{a}' = 0

  • Option 2)

     

    a{b}'+b{c}'+1=0

  • Option 3)

     

    a{a}'+c+{c}'=0

  • Option 4)

     

    b{b}'+c{c}'+1=0

Answers (1)

best_answer

 

Ange between two lines in terms of direction cosines and direction ratios -

(i)    if two lines are parallel then

        l_{1}=l_{2}, m_{1}=m_{2}, n_{1}=n_{2} or

        \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

(ii)     if two lines are perpendicular then

        l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}= 0 or1

        a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}= 0

 

-

 

Equation of lines are 

\\x = ay + b \\ z = cy + d \\\Rightarrow\frac{x-b}{a} = \frac{y}{1} = \frac{z-d}{c}

and, lines

\\x = a'y + b' \\ y = c'y + d' \\\Rightarrow\frac{x-b'}{a'} = \frac{y - d'}{c'} = \frac{z}{1}

Given that both the lines are perpendicular from the concept

aa' + c' + c = 0


Option 1)

 

c{c}'+a+{a}' = 0

Option 2)

 

a{b}'+b{c}'+1=0

Option 3)

 

a{a}'+c+{c}'=0

Option 4)

 

b{b}'+c{c}'+1=0

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