Q. 164.7( 7 Votes )

Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line

Answer :

Given: Equation of line is


To find: coordinates of foot of the perpendicular from (2, -1, 5) to the line. And find the length of the perpendicular.


Formula Used:


1. Equation of a line is


Cartesian form:


where is a point on the line and b1 : b2 : b3 is the direction ratios of the line.


2. Distance between two points (x1, y1, z1) and (x2, y2, z2) is



Explanation:


Let



So the foot of the perpendicular is (10λ + 11, -4λ - 2, -11λ - 8)


Direction ratio of the line is 10 : -4 : -11


Direction ratio of the perpendicular is


(10λ + 11 - 2) : (-4λ - 2 + 1) : (-11λ - 8 - 5)


(10λ + 9) : (-4λ - 1) : (-11λ - 13)


Since this is perpendicular to the line,


10(10λ + 9) - 4(-4λ - 1) - 11(-11λ - 13) = 0


100λ + 90 + 16λ + 4 + 121λ + 143 = 0


237λ = -237


λ = -1


So the foot of the perpendicular is (1, 2, 3)


Distance



= √14 units


Therefore, the foot of the perpendicular is (1, 2, 3) and length of perpendicular is √14 units.


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