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Question

# The coordinates of two points on the circle x2+y2âˆ’12xâˆ’16y+75=0, one nearest to the origin and the other farthest from it, are

A
(3,4)
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B
(3,2)
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C
(9,12)
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D
(9,12)
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Solution

## The correct options are B (3,4) D (9,12)The equation of the circle is x2+y2−12x−16y+75=0.Join its centre C with O.Let OC meets the circle in A and B. Centre of the circle is C≡(6,8) and radius AC=√36+64−75=5.Also, OC=√36+64=10∴OA=OC−AC=5.Now, the point on the circle nearest to the origin is A and farthest from it is B.∵OA:AC=5:5=1:1⇒A is the mid-point of OC.∴ the coordinates of A are (0+62,0+82) i.e., (3,4)Let the coordinate of B be (h,k).Since C is the mid point of AB∴3+h2=6,4+k2=8⇒h=9,k=12.∴ the coordinates are of B are (9,12).

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