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UPSC NDA 2 2016 Mathematics Solved Paper

Question Numbers: 63-64

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 plane x + y + z = 1 and 2x – y - 2z = 3.

Question : 64 of 120

Marks: +1, -0

What is the equation of the line L?

Solution:

Concept:

The symmetric form of a line in 3D is

\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}

, where

(x_1,y_1,z_1)

is a point on the line and

(a,b,c)

are direction ratios.

Explanation:

The line L passes through the point P(5, -6, 7) and has direction ratios (1, -4, 3).

Substitute into the symmetric form:

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

Multiplying all denominators by -1 gives an equivalent equation:

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

This form directly matches option A.

Answer:

Option A:

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

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