x^2(x^2+y^2)=7y^2

Question

Convert this equation into polar coordinates. Express as a function of t. Assume that r>0.

r= ANSWER

For my final answer I got r^3=7tan^2(t) which then I put r=(7tan^2(t))^(1/3)

but its wrong :,(

Answer 1

In this problem we have to keep in mind that : x= r cos (θ) and y= r sin (θ), let go ahead and substitute these values in the given rectangular equation x^2 (x^2 +y^2)= 7y^2

1. (r cos(θ))^2 [(r cos(θ))^2 + (r sin(θ)]^2. = 7 (r sin (θ))^2 [. ] brackets
2. Let work inside the brackets: r^2 cos^2(θ) [r^2 cos^2 (θ). + r^2 sin^2 (θ)] = 7 r^2 sin^2 (θ)
3. r^2 cos^2 (θ) [r^2 ( cos^2 (θ) + sin^2(θ))]. = 7 r^2 sin^2 (θ). common factor inside the brackets (r^2)
4. Applying trigonometric identities cos^2(θ) + sin^2 (θ) =1. inside the brackets
5. r^2 cos^2 (θ) [ r^2 ( 1) ] = 7 r^2 sin^2 (θ)
6. Finding the product to the left of the equation I got: r^4 cos^2(θ) = 7 r^2 sin^2(θ)
7. Dividing both sides by r^2 we get:
8. r^2 cos^2 (θ) = 7sin^2 (θ)
9. Dividing bth sides by cos^2 (θ), yield r^2 =. (7 sin^2(θ)) /( cos^2 (θ)).
10. sin(θ) / cos(θ) = tan (θ). so r^2 = 7 tan^2(θ)
11. Isolating r resulting int he square root of √ 7 tan^2(θ) = √7 tan(θ)
12. Therefore your final answer should be r = √7 tan(θ) and r=- √7 tan (θ).
13. In order to check your answer(s) graphically in a ti-84 graphing calculator graph your answer in polar form by pressing mode, change from function to polar, then r= √7 tan(θ) and -√7 tan(θ),which produce the same graph in polar for. To graph in rectangular form isolate y = +- √ x^4 /(7-x^2).

I hope that help a little bit.. if you need more help, please contact me anytime.

Answer 2

x^2(x^2+y^2)=7y^2

x^2+y^2=r^2

x^2r^2=7y^2

(rcost)^2r^2=7(rsint)^2

r^2cos^2t(r^2)=7r^2sin^2t

divide both sides by r^2

r^2cos^2t=7sin^2t

divide both sides by cos^2t

r^2=7sin^2t/cos^2t

r^2=7tan^2t

take the square root of both sides

√r^2=√(7tan^2t)

r=(√7)tant