Find coordinates of foot of perpendicular drawn from a point a(1,8,4) to the line joining the points b(0,-1,3) and c(2,-3,-1)
Answers
Answer:
Step-by-step explanation:
Step 1:
The cartesian equation of a line passing through the points B(0,−1,3)B(0,−1,3) and C(2,−3,−1)C(2,−3,−1) is x−x1x2−x1=y−y1y2−y1=z−z1z2−z1x−x1x2−x1=y−y1y2−y1=z−z1z2−z1
Now substituting the values we get
x−02−0=y+1−3+1=z−3−1−3x−02−0=y+1−3+1=z−3−1−3
⇒x2=x+1−2=z−3−4
Step 2:
Let LL be the foot of the perpendicular from the point A(1,8,4)A(1,8,4) to the given line.
A
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B L C
The coordinates of the point LL on the line BC is given by
x2=x+1−2=z−3−4x2=x+1−2=z−3−4=λ=λ
(i.e) x=2λ,y=−2λ−1,z=−4λ+3
Step 3:
The direction ratios of AL is (x2−x1),(y2−y1),(z2−z1)(x2−x1),(y2−y1),(z2−z1)
(i.e) (2λ−1),(−2λ−1−8),(−4λ+3−4)(2λ−1),(−2λ−1−8),(−4λ+3−4)
(i.e) (2λ−1),(−2λ−9),(−4λ−1)(2λ−1),(−2λ−9),(−4λ−1)
Direction ratios of the given lines are proportional to (2,−2,−4)(2,−2,−4)
Since ALAL is perpendicular to the given line BC
∴a1a2+b1b2+c1c2=0∴a1a2+b1b2+c1c2=0
(i.e) 2(2λ−1)+(−2)(−2λ−9)+(−4)(−4λ−1)=0
Step 4:
On simplifying we get,
24λ=2024λ=20
λ=2024λ=2024
λ=56λ=56
Step 5:
Hence the coordinates are
x=2(56)x=2(56)
y=−2(56)y=−2(56)−1−1
Z=−4(56)Z=−4(56)+3+3
⇒x=53⇒x=53,y=−83,,y=−83,,z=−13