Distance of a point 1,0,-3 from the planex-y-z measured parllel to the line x-2/2=y+2/3=z-6/-6

Asked by Madhureddymamilla | 6th Jan, 2020, 01:11: PM

Expert Answer:

To find the distance of a point (1,0,-3) from the plane x-y-z=9 measured along a line parallel to the line
x-2/2=y+2/3=z-6/-6
 
The given plane is x-y-z=9
So, the direction ratio of the line from the point P(1, 0, -3) to the given plane will be same as that of the given line.
The direction ratio will be (2, 3, -6)
Let this line from point P(1, 0, -3) meets the plane at Q
So, the equation of line passing through (1, 0, -3) with DR's (2, 3, -6) is

fraction numerator x minus 1 over denominator 2 end fraction equals fraction numerator y minus 0 over denominator 3 end fraction equals fraction numerator z plus 3 over denominator negative 6 end fraction equals lambda space... space left parenthesis i right parenthesis
rightwards double arrow x equals 2 lambda plus 1 comma space y equals 3 lambda comma space z equals negative 6 lambda minus 3
therefore space c o o r d i n a t e s space o f space a n y space p o i n t space o n space t h e space l i n e space a r e space 2 lambda plus 1 comma space 3 lambda comma space minus 6 lambda minus 3 space
Q space l i e s space o n space t h e space l i n e space g i v e n space b y space left parenthesis i right parenthesis
rightwards double arrow c o o r d i n a t e s space Q space w i l l space b e space left parenthesis 2 lambda plus 1 comma space 3 lambda comma space minus 6 lambda minus 3 right parenthesis space f o r space s o m e space lambda
Q space a l s o space l i e s space i n space t h e space p l a n e comma space s o space i t space b e c o m e s space t h e space p o i n t space o f space i n t e r s e c t i o n space o f space a space l i n e space a n d space p l a n e
T h e r e f o r e comma space i t space s a t i s f i e s space t h e space e q u a t i o n space o f space t h e space p l a n e
rightwards double arrow 2 lambda plus 1 minus 3 lambda plus 6 lambda plus 3 equals 9
rightwards double arrow 5 lambda plus 4 equals 9
rightwards double arrow lambda equals 1
H e n c e comma space t h e space c o o r d i n a t e s space o f space Q space a r e space left parenthesis 3 comma space 3 comma space minus 9 right parenthesis
therefore space P Q equals square root of open parentheses 3 minus 1 close parentheses squared plus open parentheses 3 minus 0 close parentheses squared plus open parentheses negative 9 plus 3 close parentheses squared end root
therefore space P Q equals square root of 4 plus 9 plus 36 end root equals 7

Answered by Renu Varma | 8th Jan, 2020, 10:22: AM