Let 0-2 and x-Xcos - -Ysin - -y-Xsin - -Ycos - such that x2-4xy-y2-aX2-bY2- where a-b are constants- Then

The correct options are B θ =π/4 C a=3, b= 1 x^2 + 4xy + y^2 = (x + y)^2 + 2xy = [(Xcosθ + Ysinθ) + (Xsinθ Ycosθ)]^2 + 2(Xcosθ + Ysinθ).( ...

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Standard XII

Mathematics

Implicit Differentiation

Let 0≤θ≤π/2...

Question

Let $0 \leq θ \leq \frac{π}{2}$ and $x = X c o s θ + Y s i n θ, y = X s i n θ - Y c o s θ$ such that
$x^{2} + 4 x y + y^{2} = a X^{2} + b Y^{2},$ where $a, b$ are constants. Then

a = - 1, b = 3

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θ = \frac{π}{4}

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a = 3, b = - 1

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θ = \frac{π}{3}

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Solution

The correct options are
B $θ = \frac{π}{4}$
C $a = 3, b = - 1$
$x^{2} + 4 x y + y^{2} = (x + y)^{2} + 2 x y$
$= [(X c o s θ + Y s i n θ) + (X s i n θ - Y c o s θ)]^{2} + 2 (X c o s θ + Y s i n θ) . (X s i n θ - Y c o s θ)$
$= [(X - Y) c o s θ + (X + Y) s i n θ]^{2} + 2. X Y (s i n^{2} θ - c o s^{2} θ) + 2 (X^{2} - Y^{2}) s i n θ . c o s θ$
$= X^{2} + Y^{2} + 4 (X^{2} - Y^{2}) s i n θ . c o s θ + 2. X Y (s i n^{2} θ - c o s^{2} θ)$
$= X^{2} (1 + 4 c o s θ . s i n θ) + Y^{2} (1 - 4 s i n θ . c o s θ) + 2. X Y (s i n^{2} θ - c o s^{2} θ)$
$= a X^{2} + b Y^{2}$
comparing coefficient,
$c o s^{2} θ = s i n^{2} θ \Rightarrow t a n^{2} θ = 1 \Rightarrow θ = \frac{π}{4}$
and $a = 3, b = - 1$
Hence, options 'B' and 'C' are correct.

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Let 0≤θ≤π2 and x=Xcosθ+Ysinθ,y=Xsinθ−Ycosθ such thatx2+4xy+y2=aX2+bY2, where a,b are constants. Then

Let $0 \leq θ \leq \frac{π}{2}$ and $x = X c o s θ + Y s i n θ, y = X s i n θ - Y c o s θ$ such that
$x^{2} + 4 x y + y^{2} = a X^{2} + b Y^{2},$ where $a, b$ are constants. Then