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UPSC NDA I 2026 Mathematics Paper

Question Numbers: 55-56

For the next two (02) items that follow:

A line L passing through the point

(-1,2,-3)

is perpendicular to the plane

P

given by

2 x+3 y+z+5=0

Question : 55 of 120

Marks: +1, -0

What is the equation of the line L ?

Solution:

Concept:

A line perpendicular to a plane has its direction vector equal to the normal vector of the plane.

Explanation:

The plane is

2x+3y+z+5=0

, so its normal vector is

(2,3,1)

Thus, the direction ratios of line

L

are

(2,3,1)

Since

L

passes through

(-1,2,-3)

, its symmetric equation is:

\frac{x+1}{2}=\frac{y-2}{3}=\frac{z+3}{1}

Let this common value be

k

. Then:

x+1=2k,\quad y-2=3k,\quad z+3=k

Multiplying by

3,2,6

respectively to make ratios equal:

3(x+1)=2(y-2)=6(z+3)

3x+3=2y-4=6z+18

Answer:

Option C:

3x+3=2y-4=6z+18

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