The graph of r = a sin(n*theta) and r = a cos(n*theta), for a is not equal 0, are are called rose curves. If n is even, the rose has 2n petals. If n is odd, the rose has n petals and a is the length of the petal. The rectangular equation x^4 + 2x^2*y^2 - 18yx^2+6y^3+ y^4 = 0 is a rose with how many petals and what is the length of the petals?
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To express the equation x^4 + 2x^2*y^2 - 18yx^2 + 6y^3 + y^4 = 0 in polar coordinates, we substitute x = r*cos(theta) and y = r*sin(theta).
Then, the equation becomes:
(r*cos(theta))^4 + 2(r*cos(theta))^2*(r*sin(theta))^2 - 18*(r*sin(theta))*(r*cos(theta))^2 + 6*(r*sin(theta))^3 + (r*sin(theta))^4 = 0
Simplifying:
r^4*cos^4(theta) + 2r^2*cos^2(theta)*sin^2(theta) - 18r^3*sin(theta)*cos^2(theta) + 6r^3*sin^3(theta) + r^4*sin^4(theta) = 0
Dividing through by r^4:
cos^4(theta) + 2cos^2(theta)*sin^2(theta)/r^2 - 18*sin(theta)*cos^2(theta)/r + 6*sin^3(theta)/r^3 + sin^4(theta) = 0
As we can see, this equation does not have the form r = a*sin(n*theta) or r = a*cos(n*theta). Therefore, it is not a rose curve.
The equation x^4 + 2x^2*y^2 - 18yx^2 + 6y^3 + y^4 = 0 does not represent any specific number of petals or petal length.
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