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Question
Find the coordinates of the centre and radius of the following circle:
1/2 (x2 + y2) + x cos θ + y sin θ − 4 = 0
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Solution
Given:
The equation of the circle is,
1/2 (x2 + y2) + x cos θ + y sin θ − 4 = 0
(Multiply by 2 we get)
x2 + y2 + 2x cos θ + 2y sin θ – 8 = 0
By comparing with the equation x2 + y2 + 2ax + 2by + c = 0
Centre = (−a, −b)
= [(−2 cos θ)/2, (−2 sin θ)/2]
= (−cos θ, −sin θ)
Radius = √(a2 + b2 − c)
= √[(−cos θ)2 + (sin θ)2 −(−8)]
= √[cos2θ + sin2θ + 8]
= √[1 + 8]
= √[9]
= 3
∴ The centre and radius of the circle is (−cos θ, −sin θ) and 3.
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