Parametric Equation of a Circle

The following is the equation of a circle in parametric form 😡 = h + r cosθy = k + r sinθwhere θ is the parameter, (h, k) is the centre and r is the radius of the circle.

Converting Rectangular Form to Parametric Form
Case (i) :Consider the rectangular equation of a circle in standard form :(x – h)2 + (y – k)2 = r2When the rectangular equation of a circle is given in standard form, we can find the center (h, k) and radius r of the circle easily. Once we get the center (h, k) and radius r, we can substitute the values for h, k and r into the equation of the circle in parametric form given above.Case (ii) :Consider the rectangular equation of a circle in general form :x2 + y2 + 2gx + 2fy + c = 0When the rectangular equation of a circle is given in general form. We can use the formulas given below to find the center (h, k) and radius r.Center (h, k) = (-g, -f)

Video Lesson

Solved Problems
In each case, convert the given rectangular equation of the circle to parametric form.Problem 1 :x2 + y2 = 36Solution :The given rectangular equation of the circle is in standard form. It can be written in the exact form (x – h)2 + (y – k)2 = r2Then, we have(x – 0)2 + (y – 0)2 = 62In the above rectangular equation of the circle,center (h, k) = (0, 0)radius r = 6Equation of a circle in parametric form 😡 = h + r cosθy = k + r sinθSubstitute h = 0, k = 0 and r = 6.x = 0 + 6 cosθ —-> x = 6 cosθy = 0 + 6 sinθ —-> y = 6 sinθProblem 2 :(x + 2)2 + (y – 3)2 = 49Solution :The given rectangular equation is in standard form and it can be written as [x – (-2)]2 + (y – 3)2 = 72In the above rectangular equation of the circle,center (h, k) = (-2, 3)radius r = 7Equation of a circle in parametric form 😡 = h + r cosθy = k + r sinθSubstitute h = -2, k = 3 and r = 7.x = -2 + 7 cosθy = 3 + 7 sinθProblem 3 :x2 + (y + 7)2 = 10Solution :The given rectangular equation is in standard form and it can be written as (x – 0)2 + [y – (-7)]2 = (√10)2In the above rectangular equation of the circle,center (h, k) = (0, -7)radius r = √10Equation of a circle in parametric form 😡 = h + r cosθy = k + r sinθSubstitute h = 0, k = -7 and r = √10.x = √10 cosθy = -7 + √10 sinθProblem 4 :x2 + y2 + 8x – 10y + 5 = 0Solution :The given rectangular equation is in general form. So, we have to find the center and radius.Comparing the given equation of the circle withx2 + y2 + 2gx + 2fy + c = 0,we get

Center : (h, k) = (-g, -f)= (-4, 5)Radius :

Equation of a circle in parametric form 😡 = h + r cosθy = k + r sinθSubstitute h = -4, k = 5 and r = 6.x = -4 + 6 cosθy = 5 + 6 sinθ

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