Coordinates of a point (x, y) on a circle having centre at \((x_{c}, y_{c})\) and polar coordinates r and θ may be expressed in parametric polar form as :
MH SET - 2019, SET Exam - 2019,
Answer: A
The parametric form is \(x = x_{c} + r\cos\theta,\ y = y_{c} + r\sin\theta\).
A point at distance r from the centre at angle \(\theta\) measured from the positive x-axis lies \(r\cos\theta\) to the right of the centre and \(r\sin\theta\) above it. Adding the centre's coordinates gives the point. As \(\theta\) runs from 0 to \(2\pi\), the point traces the whole circle.
Swapping sin and cos, as in (B), traces the same circle starting from the top and going the other way, which is not the standard polar form. Options (C) and (D) give x and y the same function of \(\theta\), so they trace a straight diagonal segment, not a circle.
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