If the transformed equation of xy-c2 when the axis are rotated through an angle of -4 in the anti clockwise direction- is pX2-qY2-rc2- then p-q-r is equal to

Original equation is xy=c^2 , θ =π4 x=Xcosθ Y sinθ=X Y√(2) y=Xsinθ +Y cosθ=X+Y√(2) Transformed equation is nbsp; (X Y√(2)) (X+Y√(2))=c^2 nbsp; ⇒ X^2 Y^2= ...

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Standard XII

Mathematics

Rotation of Axes

If the transf...

Question

If the transformed equation of $x y = c^{2}$ when the axis are rotated through an angle of $\frac{π}{4}$ (in the anti clockwise direction), is $p X^{2} + q Y^{2} = r c^{2}$ , then $p - q + r$ is equal to

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Solution

The correct option is D $4$
Original equation is $x y = c^{2}, θ = \frac{π}{4}$
$x = X cos θ - Y sin θ = \frac{X - Y}{\sqrt{2}} y = X sin θ + Y cos θ = \frac{X + Y}{\sqrt{2}}$
Transformed equation is
$(\frac{X - Y}{\sqrt{2}}) (\frac{X + Y}{\sqrt{2}}) = c^{2} \Rightarrow X^{2} - Y^{2} = 2 c^{2}$
Comparing it with the given equation
$\Rightarrow p = 1, q = - 1, r = 2$
$∴ p - q + r = 4$

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If the transformed equation of xy=c2 when the axis are rotated through an angle of π4 (in the anti clockwise direction), is pX2+qY2=rc2, then p−q+r is equal to

The correct option is D 4Original equation is xy=c2,θ=π4 x=Xcosθ−Ysinθ=X−Y√2y=Xsinθ+Ycosθ=X+Y√2 Transformed equation is (X−Y√2)(X+Y√2)=c2⇒X2−Y2=2c2 Comparing it with the given equation ⇒p=1,q=−1,r=2 ∴p−q+r=4

If the transformed equation of $x y = c^{2}$ when the axis are rotated through an angle of $\frac{π}{4}$ (in the anti clockwise direction), is $p X^{2} + q Y^{2} = r c^{2}$ , then $p - q + r$ is equal to