Question : If $x$ and $y$ are real numbers, then the least possible value of $4 (x - 2)^{2} + (y - 3)^{2} - 2 (x - 3)^{2}$ is:
Option 1: 3
Option 2: –4
Option 3: 1
Option 4: –8

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Question : If $x^{4} + x^{2} y^{2} + y^{4} = 21$ and $x^{2} + x y + y^{2} = 3$ , then what is the value of $4 x y$ ?

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Answer 1

Correct Answer: –4

Solution : Given: $x$ and $y$ are real numbers.
The expression is $4 (x - 2)^{2} + (y - 3)^{2} - 2 (x - 3)^{2}$
Use the algebraic identity, $(a - b)^{2} = a^{2} + b^{2} - 2 a b$
For the least possible value, substitute $(y - 3)^{2} = 0$
⇒ $4 (x - 2)^{2} + 0 - 2 (x - 3)^{2}$
$= 4 (x^{2} + 4 - 2 x) - 2 (x^{2} + 9 - 6 x)$
$= 4 x^{2} + 16 - 8 x - 2 x^{2} - 18 + 12 x$
$= 2 (x^{2} - 1 - 2 x)$
$= 2 (x^{2} + 1 - 2 x - 2)$
$= 2 [(x - 1)^{2} - 2]$
For the least possible value, substitute $(x - 1)^{2} = 0$ .
⇒ $2 \times - 2 = - 4$
Hence, the correct answer is –4.

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Question : If x and y are real numbers, then the least possible value of 4(x−2)2+(y−3)2−2(x−3)2 is:Option 1: 3Option 2: –4Option 3: 1Option 4: –8

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Question : If $x$ and $y$ are real numbers, then the least possible value of $4 (x - 2)^{2} + (y - 3)^{2} - 2 (x - 3)^{2}$ is:
Option 1: 3
Option 2: –4
Option 3: 1
Option 4: –8