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Question
If x = h + a cos θ, y = k + a sin θ, prove that: (x − h)2 + (y − k)2 = a2
Theorem
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Solution
x = h + a cos θ
⇒ x − h = a cos θ
y = k + a sin θ
⇒ y − k = a sin θ
Square both rearranged equations:
(x − h)2 = (a cos θ)2
(x − h)2 = a2 cos2 θ .....(1)
(y − k)2 = (a sin θ)2
(y − k)2 = a2 sin2 θ .....(2)
Add Equations 1 and 2:
(x − h)2 + (y − k)2 = a2 cos2 θ + a2 sin2 θ
(x − h)2 + (y − k)2 = a2 (cos2 θ + sin2 θ)
Use the fundamental identity sin2 θ + cos2 θ = 1
(x − h)2 + (y − k)2 = a2 (1)
(x − h)2 + (y − k)2 = a2
Hence Proved.
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