Question - A circle has the equation:
$$(x-3)^2 + (y+1)^2 = 16$$
Find parametric equations to describe the circle given that:
(a) $x = 3 + 4\cos t$
(b) $x = 3 - 4\sin t$
Much appreciated if explanation can also be given! How can coefficient be negative when it relates to radius? Also why the $x$ part is $\sin$ when it’s usually always $\cos$?
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Hint: Just plug your expression for $x$ into the original equation for the circle and solve for $y$.
For your second question, it doesn't matter which you choose for $\cos$, as long as together they satisfy the original equation. Using a different parameterization just means the point may not move around the circle in the "usual" way. The "usual" parameterization has the point start at the right side of the circle and move once around the circle counterclockwise. The direction, speed, and starting point can be varied by changing the parameterization.
(a) for given x, $y=-1+4\sin t.$ (b) for given x, $y=-1+4\cos t.$ Polar co-ordinate parametrization must be $x=rcos t, y=rsin t. $ (b) is another such parametrization which satisfies the equation of the given circle.