If x - Xcos-Ysin- and y - X sin- - Y cos- and x2 - 4xy - y2 - AX2 - BY2-0 - - - -2- then

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Domain and Range of Trigonometric Ratios

If x = Xco...

Question

If x = $X cos θ - Y sin θ$ and y = $X sin θ + Y cos θ$ and $x^{2} + 4 x y + y^{2}$ = $A X^{2} + B Y^{2}$ ,0 $\leq$ $θ$ $\leq$ $π / 2$ , then

θ

π / 6

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θ

π / 4

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A =

3

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B =

1

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Solution

The correct options are
B $θ$ = $π / 4$
C A = $3$
On putting for x and y in the given relation, we get

$(1 + 2 sin 2 θ) X^{2} + 4 cos 2 θ X Y + (1 - 2 sin 2 θ) Y^{2} = A X^{2} + B Y^{2}$ .

Comparing, we get $4 cos 2 θ$ = 0.

$∴$ 2 $θ$ = $\frac{π}{2}$ or $θ$ = $\frac{π}{4}$

$∴$ $1 + 2 sin 2 θ = A$ or $1 + 2.1 = A = 3$

$∵$ 2 $θ$ = $\frac{π}{2}$

or $1 - 2.1 = B$

$∴$ $B = - 1$

(d) is not corrrect.

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If x = Xcosθ−Ysinθ and y = Xsinθ+Ycosθ and x2+4xy+y2 = AX2+BY2,0≤ θ ≤ π/2, then

The correct options are B θ = π/4 C A =3On putting for x and y in the given relation, we get(1+2sin2θ)X2+4cos2θXY+(1−2sin2θ)Y2=AX2+BY2.Comparing, we get 4cos2θ = 0.∴ 2θ = π2 or θ = π4∴ 1+2sin2θ=A or 1+2.1=A=3∵ 2θ = π2or 1−2.1=B ∴ B=−1(d) is not corrrect.

If x = $X cos θ - Y sin θ$ and y = $X sin θ + Y cos θ$ and $x^{2} + 4 x y + y^{2}$ = $A X^{2} + B Y^{2}$ ,0 $\leq$ $θ$ $\leq$ $π / 2$ , then