Math  /  Algebra

Question2) The equation of the leminscate' r2=9cos2θr^{2}=9 \cos 2 \theta is equivalent to the rectangular equation : A) (x2+y2)2=9\left(x^{2}+y^{2}\right)^{2}=9 C) (x2+y2)2=9(x2y2)2\left(x^{2}+y^{2}\right)^{2}=9\left(x^{2}-y^{2}\right)^{2} B) (x2+y2)2=9(x2y2)\left(x^{2}+y^{2}\right)^{2}=9\left(x^{2}-y^{2}\right) D) (x2+y2)=9(x2y2)\left(x^{2}+y^{2}\right)=9\left(x^{2}-y^{2}\right)

Studdy Solution

STEP 1

1. The given equation is in polar coordinates: r2=9cos2θ r^2 = 9 \cos 2\theta .
2. We need to convert this polar equation to a rectangular (Cartesian) equation.
3. The relationships between polar and rectangular coordinates are x=rcosθ x = r \cos \theta , y=rsinθ y = r \sin \theta , and r2=x2+y2 r^2 = x^2 + y^2 .

STEP 2

1. Express cos2θ\cos 2\theta in terms of cosθ\cos \theta and sinθ\sin \theta.
2. Substitute cosθ\cos \theta and sinθ\sin \theta with xx and yy using the polar to rectangular conversion.
3. Substitute r2r^2 with x2+y2x^2 + y^2.
4. Simplify the equation to match one of the given options.

STEP 3

Use the double-angle identity for cosine: cos2θ=cos2θsin2θ\cos 2\theta = \cos^2 \theta - \sin^2 \theta.
cos2θ=cos2θsin2θ \cos 2\theta = \cos^2 \theta - \sin^2 \theta

STEP 4

Substitute cosθ=xr\cos \theta = \frac{x}{r} and sinθ=yr\sin \theta = \frac{y}{r} into the double-angle identity.
cos2θ=(xr)2(yr)2 \cos 2\theta = \left(\frac{x}{r}\right)^2 - \left(\frac{y}{r}\right)^2

STEP 5

Simplify the expression for cos2θ\cos 2\theta.
cos2θ=x2r2y2r2=x2y2r2 \cos 2\theta = \frac{x^2}{r^2} - \frac{y^2}{r^2} = \frac{x^2 - y^2}{r^2}

STEP 6

Substitute cos2θ\cos 2\theta and r2r^2 into the original equation r2=9cos2θ r^2 = 9 \cos 2\theta .
r2=9(x2y2r2) r^2 = 9 \left(\frac{x^2 - y^2}{r^2}\right)

STEP 7

Multiply both sides by r2r^2 to clear the fraction.
r4=9(x2y2) r^4 = 9 (x^2 - y^2)

STEP 8

Recognize that r2=x2+y2r^2 = x^2 + y^2. Therefore, r4=(x2+y2)2r^4 = (x^2 + y^2)^2.
(x2+y2)2=9(x2y2) (x^2 + y^2)^2 = 9 (x^2 - y^2)

STEP 9

Compare the derived equation with the given options to identify the correct choice.
The correct equation is:
(x2+y2)2=9(x2y2) (x^2 + y^2)^2 = 9 (x^2 - y^2) This corresponds to option B.
Solution: The rectangular equation equivalent to r2=9cos2θ r^2 = 9 \cos 2\theta is (x2+y2)2=9(x2y2)\boxed{(x^2 + y^2)^2 = 9 (x^2 - y^2)}.

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