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Find the equations of the line passing through the point (3, 0, 1) and parallel to the planes   x + 2 y = 0 and 3 y − z = 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  point  is(3, 0, 1)  and  the  equation  of  planes  are                                                     x + 2 y = 0                                                                               … ( i ) a n d                                       3 y − z = 0                                                                             … ( i i ) Equation  of  any  line  l  passing  through  (3, 0, 1)  is l :     x − 3 a = y − 0 b = z − 1 c D i r e c t i o n     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 )     a n d     ( i i )     a r e     ( 1 ,   2 ,   0 )     a n d     ( 0 ,   3 ,   − 1 ) Since,  the  line  is  parallel  to  both  the  planes. ∴                       1 . a + 2 . b + 0 . c = 0         ⇒ a + 2 b + 0 c = 0 a n d           0 . a + 3 . b − 1 . c = 0         ⇒ 0 . a + 3 b − c = 0 S o ,     a − 2 − 0 = − b − 1 − 0 = c 3 − 0 = λ ∴     a = − 2 λ ,     b = λ ,     c = 3 λ S o ,     e q u a t i o n     o f     l i n e     i s                 x − 3 − 2 λ = y − 0 λ = z − 1 3 λ H e n c e ,     t h e     r e q u i r e d     e q u a t i o n     x − 3 − 2 + y − 0 1 + z − 1 3 o r     i n     v e c t o r     f o r m     i s     ( x − 3 ) i ^ + y j ^ + ( z − 1 ) k ^ = λ ( − 2 i ^ + j ^ + 3 k ^ )

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