If the centre of a circle is (2a, a-7), then the values of 'a', if the circle passes through the point (11,-9) and has diameter 10√2 units will be:
5 and 3
By given condition,
Distance between the centre C(2a, a-7) and the point P(11,-9), which lie on the circle = Radius of the circle
∴ Radius of circle = √(11−2a)2+(−9−a+7)2 ...(i)
∵ Distance between the points (x1,y1) and (x2,y2)
d=√(x2−x1)2+(x2−y1)2
Given that, length of diameter = 10√2
Length of radius = Length of diameter2
=10√22=5√2
Putting this value in equation (i), we get
50=(11−2a)2+(2+a)2⇒50=121+4a2−44a+4+a2+4a⇒5a2−40a+75=0⇒a2−8a+15=0⇒a2−5a−3a+15=0 [by factorization method]
⇒a(a−5)−3(a−5)=0⇒(a−5)(a−3)=0∴a=3,5
Hence, the required values of 'a' are 5 and 3.