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Question 14
If the centre of a circle is (2a, a-7), then find the values of a , if the circle passes through the point (11,-9) and has diameter 102 units.

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Solution

By given condition,

Distance between the centre C(2a , a-7) and the point P(11,-9), which lie on the circle = Radius of circle
Radius of circle=(112a)2+(9a+7)2 ...(i)[distance between the points(x1,y1) and (x2,y2)d=(x2x1)2+(x2y1)2]Given that, length of diameter =102Length of radius =Length of diameter2=10222=52Put this value in Eq.(i), we get50=(112a)2+(2+a)250=121+4a244a+4+a2+4a5a240a+75=0a28a+15=0a25a3a+15=0 [by splitting the middle term]a(a5)3(a5)=0(a5)(a3)=0a=3,5Hence, the required values of a are 5 and 3.

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