$What does the equation (x - a)^{2} + (y - b)^{2} = r^{2} become when the axes are transferred to parallel axes through the point (a-c,b) ?$

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Solution

$W e h a v e . (x - a)^{2} + (y - b)^{2} = r^{2} . . . . (i) s u b s t i t u t i n g x = X + (a - c), y = Y + b i n t h e e q u a t i o n (I), w e g e t {[X + a - c - a^{2}]}^{2} + {[Y + b - b]}^{2} \Rightarrow [X - c]^{2} + [Y]^{2} = r^{2} \Rightarrow X^{2} + c^{2} - 2 x c + Y^{2} = r^{2} \Rightarrow X^{2} + Y^{2} - 2 c x = r^{2} - c^{2} H e n c e, t h e r e q u i r e d e q u a t i o n i s x^{2} + y^{2} - 2 c x = r^{2} - c^{2} \Rightarrow X^{2} + Y^{2} - 2 c x = r^{2} - c^{2}$

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