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If $x = 3 + 2 3 − 2$ and $y = 3 − 2 3 + 2$ , then what is the value of $x 2 + y 2$ ?

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100

answer is C.

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Detailed Solution

It is given that

x = 3 + 2 3 − 2 and y = 3 − 2 3 + 2

.
Multiplying the numerator and the denominator by

3 + 2

Therefore,

x = 5 + 26

…(i)
Similarly,

y = 5 - 2 \sqrt{6}

…(ii)
Squaring on both the sides and using the identity,

a + b 2 = a 2 + 2 a b + b 2

x 2 = 5 + 26 2 = 52 + 262 + 2 ⋅ 5 ⋅ 26

x 2 = 49 + 206

…(iii)
From Eq. (i),

1 x = 1 5 + 26

1 x = 1 5 + 26 = 1 5 + 26 × 5 − 26 5 − 26 = 5 − 26 52 − 262 = 5 − 26 25 − 24 = 5 − 26 1

Squaring both the sides we get,

y 2 = 5 − 26 2

y 2 = 52 + 262 − 2 × 5 × 26

y 2 = 25 + 24 − 206

y 2 = 49 − 206

…(iv)
Adding Eq. (iii) and (iv), we get

x 2 + y 2 = 49 + 206 + 49 − 206 x 2 + y 2 = 98

Therefore, the correct option is (3).

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