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Three Dimensional Geometry

Question
CBSEENMA12033510

Show that the lines:
            fraction numerator straight x minus straight a plus straight d over denominator straight alpha minus straight delta end fraction space equals space fraction numerator straight y minus straight a over denominator straight alpha end fraction space equals space fraction numerator straight z minus straight a minus straight d over denominator straight a plus straight delta end fraction

and       fraction numerator straight x minus straight b plus straight c over denominator straight beta minus straight gamma end fraction space equals space fraction numerator straight y minus straight b over denominator straight beta end fraction space equals space fraction numerator straight z minus straight b minus straight c over denominator straight beta plus straight gamma end fraction are coplanar. 

Solution
The equations of given lines are
                          fraction numerator straight x minus straight a plus straight d over denominator straight alpha minus straight delta end fraction space equals space fraction numerator straight y minus straight a over denominator straight alpha end fraction space equals space fraction numerator straight z minus straight a minus straight d over denominator straight a plus straight delta end fraction
and          fraction numerator straight x minus straight b plus straight c over denominator straight beta minus straight gamma end fraction space equals space fraction numerator straight y minus straight b over denominator straight beta end fraction space equals space fraction numerator straight z minus straight b minus straight c over denominator straight beta plus straight gamma end fraction
These lines are coplanar
    if            open vertical bar table row cell straight b minus straight c minus straight a plus straight d end cell cell straight b minus straight a end cell cell straight b plus straight c minus straight a minus straight d end cell row cell straight a minus straight delta end cell straight alpha cell straight alpha plus straight delta end cell row cell straight beta minus straight gamma end cell straight beta cell straight beta plus straight gamma end cell end table close vertical bar space equals space 0

i.e. if        open vertical bar table row cell 2 straight b minus 2 straight a end cell cell straight b minus straight a end cell cell straight b plus straight c minus straight a minus straight d end cell row cell 2 straight alpha end cell straight alpha cell straight alpha plus straight delta end cell row cell 2 straight beta end cell straight beta cell straight beta plus straight gamma end cell end table close vertical bar space equals space 0 comma space space by space straight C subscript 1 plus straight C subscript 3
i.e. if   open vertical bar table row cell straight b minus straight a end cell cell straight b minus straight a end cell cell straight b plus straight c minus straight a minus straight d end cell row straight alpha straight alpha cell straight alpha plus straight delta end cell row straight beta straight beta cell straight beta plus straight gamma end cell end table close vertical bar space equals 0             

i.e. if    2(0) = 0    [Since C1, C2 are identical]
i.e. if    0 = 0, which is true.
Hence the result.

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.