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NDA Math Three Dimensional Geometry

Consider the following for the next two (02) items that follow:A line L passes through the point

P(5,-6,7)

and is parallel to the planes

x+y+z=1

and

2x-y-2z=3

[NDA II 2016]

Question : 54 of 92

Marks: +1, -0

Solution:

Let a, b, c be the direction ratios of the line.

Then its equation is

\; \frac{x-5}{a} = \; \frac{y+6}{b} = \; \frac{z-7}{c}

...(i)

Since (i) is parallel to the planes

(x+y+z=1)

and

2x-y-2z=3

then

a(1)+b(1)+c(1)=0

and

a(z)+b(-1)+c(-2)=0

By cross multiplication

\; \frac{a}{-1} = \; \frac{b}{4} = \; \frac{c}{-3} = \lambda

\Rightarrow a = -\lambda, b = 4\lambda, c = -3\lambda

⇒ Direction ratios of,the line are

\langle -1,4,-3 \rangle = \langle 1,-4,3 \rangle

Substituting a, b, c in (i), we get

\; \frac{x-5}{-1} = \; \frac{y+6}{4} = \; \frac{z-7}{-3}

...(ii)

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