Which of the following equations represents a | vedclass

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Question

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

Vedclass · Accepted Answer

$r = 2 \sin 	heta$

Vedclass · Answer

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

Which of the following equations represents a circle?

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