QJEEM04Ma07f8: JEE Main 2004 Mathematics - Class 12 JEEM04Ma07f8 Solution: A line with direction cosines proportional to meets each of the …

Question

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A line with direction cosines proportional to

2,1,2

meets each of the lines

x=y+a=z

and

x+a=2y=2z

. The co-ordinates of each of the points of intersection are given by :

Accepted Answer

Let a point on the line

x = y + a = z

is

\left( {\lambda ,\lambda - a,\lambda } \right)

and a point on the line

x + a = 2y = 2z

is

\left( {\mu - a,{\mu \over 2},{\mu \over 2}} \right),

then direction ratio of the line joining these points are

\lambda - \mu + a,\,\,\lambda - a - {\mu \over 2},\,\,\lambda - {\mu \over 2}

If it represents the required line, then

{{\lambda - \mu + a} \over 2}

= {{\lambda - a - {\mu \over 2}} \over 1}

= {{\lambda - {\mu \over 2}} \over 2}

on solving we get

\lambda = 3a,\,\mu = 2a

\therefore

The required points of intersection are

\left( {3a,3a - a,3a} \right)

and

\left( {2a - a,{{2a} \over 2},{{2a} \over 2}} \right)

or

\left( {3a,2a,3a} \right)

and

\left( {a,a,a} \right)

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