UPSC NDA Math Model Paper 2

Given: Equation of line

x−3

1

=

y−4

2

=

z−5

1

and equation of plane is x - y + z - 5 = 0
First let's find out if they given line is parallel to the plane or not.
As we can see that, the direction ratios of the given line are: (1, 2, 1)
Similarly, the direction ratios of the normal to the given plane are: (1, -1, 1)
⇒ 1 × 1 - 2 × 1 + 1 × 1 = 0
So, the given line is parallel to the given plane.
Let's find out a point on the given line
As we can see that, point Q (3, 4, 5) lies on the line.
So, finding the distance between the point Q and the given plane is same as finding the distance between the given line and plane
As we know that, distance between a point and a plane is given by: |

Ax1+By1+Cz1−d

√A2+B2+C2

|
Here, x1=3,y1=4 and z1=5
⇒d=|

3×1+4×(−1)+5×1−5

√12+(−1)2+12

| =

1

√3

units
Hence, option B is the correct answer.