Which of the following equations represents a | vedclass

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(A) To determine which equation represents a circle,we convert the polar equations to Cartesian coordinates using $x = r \cos 	heta$ and $y = r \sin

Vedclass · Accepted Answer

$r = 2 \sin 	heta$

Vedclass · Answer

(A) To determine which equation represents a circle,we convert the polar equations to Cartesian coordinates using $x = r \cos 	heta$ and $y = r \sin 	heta$,where $r^2 = x^2 + y^2$.For option $A$: $r = 2 \sin 	heta$Multiply both sides by $r$: $r^2 = 2r \sin 	heta$Substituting $r^2 = x^2 + y^2$ and $y = r \sin 	heta$:$x^2 + y^2 = 2y$$x^2 + y^2 - 2y = 0$This is the equation of a circle with center $(0, 1)$ and radius $1$.For option $B$: $r^2 \cos 2 	heta = 1$Using $\cos 2 	heta = \cos^2 	heta - \sin^2 	heta$:$r^2(\cos^2 	heta - \sin^2 	heta) = 1$$(r \cos 	heta)^2 - (r \sin 	heta)^2 = 1$$x^2 - y^2 = 1$,which is a rectangular hyperbola.For option $C$: $r(4 \cos 	heta + 5 \sin 	heta) = 3$$4r \cos 	heta + 5r \sin 	heta = 3$$4x + 5y = 3$,which is a straight line.For option $D$: $5 = r(1 + \sqrt{2} \cos 	heta)$This is the polar form of a conic section $r = \frac{ed}{1 + e \cos 	heta}$,which represents a parabola,ellipse,or hyperbola depending on eccentricity $e$.

Which of the following equations represents a circle?

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