Three Dimensional Geometry

Question

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
3 y + 4 z – 6 = 0

Solution

The equation of plane is
0x + 3 y + 4 z = 6 ...(1)
Dividing both sides by $square root of left parenthesis 0 right parenthesis squared plus left parenthesis 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space equals space square root of 0 plus 9 plus 16 end root space equals space square root of 25 space equals space 5 comma space we space get comma$
$0 straight x plus 3 over 5 straight y plus 4 over 5 straight z space equals space 6 over 5$
It is of form lx + my + nz = p where l = 0, $straight m space equals 3 over 5 comma space straight n space equals space 4 over 5 comma space straight p space equals space 6 over 5$
∴ direction cosines of the normal OP are 0, $3 over 5 comma space 4 over 5$ where O is origin and P (x₁, y₁, z₁) is foot of perpendicular.
Direction ratios of OP are x₁– 0, y₁ – 0, z₁ – 0 i.e. x₁, y₁, z₁.
Since direction cosines and direction ratios of a line are proportional.
$therefore space space space space space space space straight x subscript 1 over 0 space equals space fraction numerator straight y subscript 1 over denominator begin display style 3 over 5 end style end fraction space equals fraction numerator straight z subscript 1 over denominator begin display style 4 over 5 end style end fraction space equals space straight k comma space space say therefore space space space space space space space space straight x subscript 1 space equals space 0 comma space space space straight y subscript 1 space equals space 3 over 5 straight k comma space space space straight z subscript 1 space equals space 4 over 5 straight k therefore space space space space straight P space is space open parentheses 0 comma space 3 over 5 straight k comma space space 4 over 5 straight k close parentheses$
Since P lies on plane (1)
$therefore space space space space 3 space open parentheses 3 over 5 straight k close parentheses space plus space 4 open parentheses 4 over 5 straight k close parentheses space equals space 6 therefore space space space space space 9 space straight k space plus space 16 space straight k space equals space 30 space space space space space or space space space 25 space straight k space equals space 30 space space space space rightwards double arrow space space space space straight k space equals space 6 over 5 therefore space space space space space straight P space is space open parentheses 0 comma space 18 over 25 comma space 24 over 25 close parentheses comma space which space is space foot space of space perpendicular. space$

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