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UPSC NDA I 2022 Mathematics Solved Paper

Question Numbers: 94-95

Consider the following for the next two (2) items that follow :

Let the plane

\frac{2x}{k} + \frac{2y}{3} + \frac{z}{3} = 2

pass through the point (2, 3, -6).

Question : 94 of 120

Marks: +1, -0

What are the direction ratios of the normal to the plane ?

Solution:

Concept:

The equation of a plane through a point

(x_1, y_1, z_1)

with normal direction ratios

(a, b, c)

a(x - x_1) + b(y - y_1) + c(z - z_1) = 0

Explanation:

Given plane:

\displaystyle \frac{2x}{k} + \frac{2y}{3} + \frac{z}{3} = 2

passes through

(2,3,-6)

Substitute the point:

\displaystyle \frac{2 \times 2}{k} + \frac{2 \times 3}{3} + \frac{(-6)}{3} = 2

Simplify:

\displaystyle \frac{4}{k} + 2 - 2 = 2 \Rightarrow \frac{4}{k} = 2 \Rightarrow k = 2

Replace

k=2

in the plane equation:

\displaystyle \frac{2x}{2} + \frac{2y}{3} + \frac{z}{3} = 2

Multiply by

3

3x + 2y + z = 6

The coefficients of

x, y, z

are the direction ratios of the normal:

\langle 3, 2, 1 \rangle

Answer:

\langle 3, 2, 1 \rangle

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