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The point on a circle nearest to the poi...

The point on a circle nearest to the point `P(2,1)` is at a distance of 4 units and the farthest point is (6, 5). Then find the equation of the circle.

Text Solution

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In the figure, point nearest to P(2,1) on the circle is A(h,k) such that AP`=4` units.
Also, point B(6,5) is on the circle which is farthest point from P. Clearly, AB is diameter of the circle.
Now, `PB=sqrt((6-2)^(2)+(5-1)^(2))=4sqrt(2)`
`:. AB=PB-PA=4(sqrt(2)-1)`
Thus, `(AB)/(AP)=(sqrt(2)-1)/(1)`
`:. A=((6-2(sqrt(2)-1))/(1+(sqrt(2)-1)),(5+1(sqrt(2)-1))/(1+(sqrt(2)-1)))`
`=(2+2sqrt(2),1+2sqrt(2))`
Hence, equation of circle having AB as diameter is
`(x-6)(x-2-2sqrt(2))+(y-5)(y-1-2sqrt(2))=0`
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