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Calculus Level 4

Let B = { ( x , y ) ∈ R 2 ; x 2 + 2 x y + y 2 ≤ 4 } B = \{(x,y) \in \mathbb{R}^2; \space x^2 + 2xy + y^2 \leq 4\}

Which of the following two pictures best represents B B ?

a)

b)

both Neither a) nor b) a) b)

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

x 2 + 2 x y + y 2 ≤ 4 ⟺ ( x + y ) 2 ≤ 4 ⟺ ∣ x + y ∣ ≤ 2 ⟺ − 2 ≤ x + y ≤ 2 ⟺ x^2 + 2xy + y^2 \leq 4 \iff (x + y)^2 \leq 4 \iff |x + y| \leq 2 \iff -2 \leq x + y \leq 2 \iff { x + y ≤ 2 ⟺ y ≤ 2 − x , and − ( x + y ) ≤ 2 ⟺ y ≥ − 2 − x \begin{cases} x + y \leq 2 \iff y \leq 2 -x, \text{ and } \\ - (x + y) \leq 2 \iff y\ge -2 - x \end{cases} Therefore, b) represents B B better than a), i. e, B B is the region between the two parallel lines y = 2 − x y = 2 -x and y = − 2 − x y = -2 - x ( both inclusive)

Example.-

( 4 , − 2 ) ∈ B , ( 3 , − 4.5 ) ∈ B . . . (4, -2) \in B, \quad (3, -4.5) \in B...

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice one! It was conceptual

Aditya Narayan Sharma - 4 years, 4 months ago

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Thank yoy very much Aditya, my π > 3 \pi > 3 !... ( :))

Guillermo Templado - 4 years, 4 months ago

Nice trap! ;-)

Andreas Wendler - 4 years, 4 months ago

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haha, I think ( :))

Guillermo Templado - 4 years, 4 months ago

Sir , shouldn't the region between the lines be shaded. Option B only represents boundaries and not the actual region

Rishi Sharma - 4 years, 4 months ago

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