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UPSC NDA I 2015 Mathematics Paper

Question Numbers: 55-56

Directions: For the next two (2) items that follow:

From the point P(3, -1, 11), a perpendicular is drawn on the line L given by the equation (x)/2 = (y - 2)/3 = (z - 3)/4 . Let Q be the foot of the perpendicular.

Question : 56 of 120

Marks: +1, -0

What is the length of the line segment PQ?

$\sqrt{47}$ units
7 units
$\sqrt{53}$ units
8 units

Validate

Solution:

Concept:

The length of a line segment is the magnitude of the direction vector.

If direction ratios are

a

b

c

, then length =

\sqrt{a^2 + b^2 + c^2}

Explanation:

From the given conditions, the direction ratios of PQ are obtained.

They are:

2(1)-3 = -1

3(1)+3 = 6

4(1)-8 = -4

Thus, the direction vector is

(-1, 6, -4)

The length of PQ is the magnitude of this vector:

\sqrt{(-1)^2 + 6^2 + (-4)^2} = \sqrt{1 + 36 + 16} = \sqrt{53}

units.

Answer:

\sqrt{53}

units (Option C).

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