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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 : 55 of 120

Marks: +1, -0

what are the direction ratios of the line segment PQ?

$\{1, 6, 4\}$
$\{-1, 6, -4\}$
$\{-1, -6, 4\}$
$\{2, -6, 4\}$

Validate

Solution:

Concept:

If two lines are perpendicular, the dot product of their direction ratios is zero.

Explanation:

Given line

L

\frac{x}{2} = \frac{y-2}{3} = \frac{z-3}{4} = \lambda

Coordinates of any point on

L

x = 2\lambda

y = 3\lambda + 2

z = 4\lambda + 3

Let

Q = (2\lambda,\ 3\lambda+2,\ 4\lambda+3)

Point

P = (3,\ -1,\ 11)

Direction ratios of

PQ

2\lambda - 3,\ (3\lambda+2) - (-1),\ (4\lambda+3) - 11

⇒

2\lambda - 3,\ 3\lambda + 3,\ 4\lambda - 8

Direction ratios of line

L

2,\ 3,\ 4

Since

PQ \perp L

, dot product =

0

2(2\lambda - 3) + 3(3\lambda + 3) + 4(4\lambda - 8) = 0

⇒

4\lambda - 6 + 9\lambda + 9 + 16\lambda - 32 = 0

⇒

29\lambda - 29 = 0

⇒

\lambda = 1

Substitute

\lambda = 1

: direction ratios of

PQ = 2(1)-3,\ 3(1)+3,\ 4(1)-8

⇒

-1,\ 6,\ -4

Answer:

\{-1, 6, -4\}

(Option B)

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