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UPSC CDS 1 2020 Math Paper

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Question : 17 of 100

Marks: +1, -0

If

x^{2} + 9y^{2} = 6xy

y : x

equal to?

1 : 3
1 : 2
2 : 1
3 : 1

Solution:

Concept:

The given equation can be rearranged into a perfect square form to find the relation between

x

and

y

.

Explanation:

Start with

x^{2} + 9y^{2} = 6xy

.

Bring all terms to one side:

x^{2} - 6xy + 9y^{2} = 0

.

Notice that

x^{2} - 6xy + 9y^{2} = (x - 3y)^{2}

because

(a - b)^{2} = a^{2} - 2ab + b^{2}

with

a = x

and

b = 3y

gives

x^{2} - 2 \cdot x \cdot 3y + (3y)^{2} = x^{2} - 6xy + 9y^{2}

.

Thus,

(x - 3y)^{2} = 0

.

Taking square root on both sides gives

x - 3y = 0

, i.e.,

x = 3y

.

Therefore,

y : x = y : 3y = 1 : 3

.

Answer:

1 : 3

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