If x- 4- 1- 2 and y- 2-2- 2 1- 2 where - is a real parameter then Z- x 2 -xy- y 2 lies between

The correct option is A [ 2,6 ] Choose λ =tanα #160; ∴ x=2sin 2α #160; ,y=2cos 2α #160; ∴ Z= x ^ 2 xy+ y ^ 2 =4.1 2sin 4α #160; ...

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Standard XII

Mathematics

Domain and Range of Basic Inverse Trigonometric Functions

If x= 4λ 1+...

Question

If $x = \frac{4 λ}{1 + λ^{2}}$ and $y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}$ where $λ$ is a real parameter then $Z = x^{2} - x y + y^{2}$ lies between

[2, 6]

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[2, 4]

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[4, 6]

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Solution

The correct option is A $[2, 6]$
Choose $λ = tan α$ $∴ x = 2 sin 2 α, y = 2 cos 2 α$
$∴ Z = x^{2} - x y + y^{2} = 4.1 - 2 sin 4 α$
Min value of $Z = 4 - 2.1 = 2$ and max. value is $4 - 2 (- 1) = 6$
$∴ Z \in [2, 6]$

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Similar questions

If $x = \frac{4 λ}{1 + λ^{2}}$ and $y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}$ where $λ$ is a real parameter and $x^{2} - x y + y^{2}$ lies between $[a, b]$ then $(a + b)$ is

Q. If

x = \frac{4 λ}{1 + λ^{2}}

and

y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}

, where

λ

is a real parameter such that

x^{2} - x y + y^{2}

lies in the interval

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, then the minimum value of

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3 cos ϕ = p^{2} - (a + b) p + 19

holds true is .

Q. If

x = \frac{4 α}{1 + α^{2}}

and

y = \frac{2 - 2 α^{2}}{1 + α^{2}},

where

α

is a real parameter, then

x^{2} - x y + y^{2}

lines between

Q. The number of distinct real values of

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for which the lines

\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{λ^{2}}

and

\frac{x - 3}{1} = \frac{y - 2}{λ^{2}} = \frac{z - 1}{2}

are coplanar is:

λ_{1}

and

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are two values of

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If x=4λ1+λ2 and y=2−2λ21+λ2 where λ is a real parameter then Z=x2−xy+y2 lies between

The correct option is A [2,6]Choose λ=tanα ∴x=2sin2α,y=2cos2α∴Z=x2−xy+y2=4.1−2sin4αMin value of Z=4−2.1=2 and max. value is 4−2(−1)=6∴Z∈[2,6]

If $x = \frac{4 λ}{1 + λ^{2}}$ and $y = \frac{2 - 2 λ^{2}}{1 + λ^{2}}$ where $λ$ is a real parameter then $Z = x^{2} - x y + y^{2}$ lies between

The correct option is A $[2, 6]$
Choose $λ = tan α$ $∴ x = 2 sin 2 α, y = 2 cos 2 α$
$∴ Z = x^{2} - x y + y^{2} = 4.1 - 2 sin 4 α$
Min value of $Z = 4 - 2.1 = 2$ and max. value is $4 - 2 (- 1) = 6$
$∴ Z \in [2, 6]$