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Question

The centre of a circle is (2a, a-7). Find the values of a if the circle passes through the point (11, -9) and has diameter 102 units.

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Solution

The distance of the centre of the circle from any point on the circle is the radius of the circle where the radius is half of the diameter.
Radius = fraction numerator 10 square root of 2 over denominator 2 end fraction ---- (i)
Radius = square root of left parenthesis 2 a minus 11 right parenthesis squared plus left parenthesis a minus 7 plus 9 right parenthesis squared end root ---- ii
From (i) and (ii)

square root of left parenthesis 2 a minus 11 right parenthesis squared plus left parenthesis a minus 7 plus 9 right parenthesis squared end root = fraction numerator 10 square root of 2 over denominator 2 end fraction
Square both sides,
4 a squared plus 121 minus 44 a plus a squared plus 4 plus 4 a equals 50 5 a squared minus 40 a plus 75 equals 0 a squared minus 8 a plus 15 equals 0 a squared minus 3 a minus 5 a plus 15 equals 0 a left parenthesis a minus 3 right parenthesis minus 5 left parenthesis a minus 3 right parenthesis equals 0 left parenthesis a minus 5 right parenthesis left parenthesis a minus 3 right parenthesis equals 0 a equals 5 comma 3


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