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in Three Dimensional Geometry by (4.0k points)
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(i) (a) A line makes equal angles with the coordinate axis. Find the direction cosines.

b) Find the equation of the Plane Passing through (1, 1, -1),(2, 3, 5) an (1, 4, -5) 

(ii) Find p and q, if the plane x + py + qz = 0 is perpendicular to the plane 3x + 2y + z = 0 and the line \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{1}\)

1 Answer

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Best answer

(i) We have;

l2 + m2 + n2 =1

cos2θ + cos2 θ + cos2 θ =1

= 3cosθ = 1

= cos2 θ = \(\frac{1}{3}\)

= cos θ = ± \(\frac{1}{\sqrt3}\)

Therefore direction cosines are 

± \(\frac{1}{\sqrt3}\) ,± \(\frac{1}{\sqrt3}\) ,± \(\frac{1}{\sqrt3}\) ,

(b) Equation of plane is

(ii) Since the Plane x + py + qz = 0 is perpendicular to the Line \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{1}\)  we have their direction ratios proportional. Plane dr’s is 1, p, q and dr’s of Line is 2, 3, 1.

⇒ \(\frac{1}{2}=\frac{p}{3}=\frac{q}{1}\\⇒p=\frac{3}{2},q=\frac{1}{2}\)

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