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UPSC NDA I 2022 Mathematics Solved Paper

Question : 81 of 120

Marks: +1, -0

\text{If the points with coordinates }(-5,0),\ (5p^{2},10p)\text{ and }(5q^{2},10q)\text{ are collinear, then what is the value of }pq\text{ where }p\neq q\text{?}

Solution:

Concept:

Three points are collinear if the area of the triangle formed by them is zero.

Area of a triangle with vertices

(x_1,y_1)

(x_2,y_2)

(x_3,y_3)

is given by

\frac{1}{2}\left[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right]

Explanation:

Let the points be

A(-5,0)

B(5p^2,10p)

, and

C(5q^2,10q)

Since they are collinear, Area of

\triangle ABC = 0

Using the area formula:

0 = \frac{1}{2}\left[-5(10p - 10q) + 5p^2(10q - 0) + 5q^2(0 - 10p)\right]

Simplify step by step:

0 = \frac{1}{2}\left[-50p + 50q + 50p^2q - 50q^2p\right]

Multiply both sides by 2:

0 = -50p + 50q + 50p^2q - 50q^2p

Rearrange:

0 = 50(q - p) + 50pq(p - q)

Factor

50(p - q)

0 = 50(p - q)(-1 + pq)

Since

p \neq q

p - q \neq 0

, thus

pq - 1 = 0

Therefore,

pq = 1

Answer:

pq = 1

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