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Find the length and the foot of the perpendicular from the point ( 1 , 3 2 , 2 ) to the plane 2 x − 2 y + 4 z + 5 = 0 .

0 2 Views | Posted 3 months ago
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    Answered by

    Payal Gupta | Contributor-Level 10

    3 months ago

    This is a long answer type question as classified in NCERT Exemplar

    Given  plane  is  2x−2y+4z+5=0  and  the  given  points  is(1, 32, 2) D ' r a t i o s     o f     t h e     n o r m a l     t o     t h e     p l a n e     a r e     2 ,   − 2 ,   4 So,  the?  equation  of  line  passing  through  (1, 32, 2)  and  whose  d'ratios  are  equal  to  the D ' r a t i o s     o f     t h e     n o r m a l     t o     t h e     p l a n e     i . e . , 2 ,   − 2 ,   4     i s     x − 1 2 = y − 3 2 − 2 = z − 2 4 = λ ∴Any  point  in  the  plane  is  2λ+1,  −2λ+32,  4λ+2 Since,  the  point  lies  in  the  plane,  then     2 ( 2 λ + 1 ) − 2 ( − 2 λ + 3 2 ) + 4 ( 4 λ + 2 ) + 5 = 0 ⇒                                             4 λ + 2 + 4 λ − 3 + 1 6 λ + 8 + 5 = 0 ⇒ 2 4 λ + 1 2 = 0                   ∴ λ = − 1 2 So,  the  coordinates  of  the  point  in  the  plane  are             2 ( − 1 2 ) + 1 ,     − 2 ( − 1 2 ) + 3 2 ,     4 ( − 1 2 ) + 2         i . e . ,     0 ,   5 2 ,   0 H e n c e ,     t h e     f o o t     o f     t h e     p e r p e n d i c u l a r     i s     ( 0 ,   5 2 ,   0 )     a n d     t h e r e q u i r e d     l e n g t h = ( 1 − 0 ) 2 + ( 3 2 − 5 2 ) 2 + ( 2 − 0 ) 2                                                                       = 1 + 1 + 4 = 6   u n i t s

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