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Question

The co-ordinates of the point on the circle x2+y212x4y+30=0 which is farthest from the origin are:

A
(9, 3)
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B
(8, 5)
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C
(12, 4)
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D
none of these
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Solution

The correct option is A (9, 3)
Given equation of circle x2+y212x4y+30=0

(x6)236+(y2)24+30=0(x6)2+(y2)2=10

Maximum distance of (0,0) from circle is
=(60)2+(20)2+10=210+10=310

Let coordinates of point P(x1,y1)

then (x10)2+(y10)2=310(1)

and this point also satisfies circle

(x16)2+(y12)2=10(2)

Solving (1)x12+y12=90(3)

Solving (2)x12+y1212x14y1+30=0(4)

Solving (3),(4)

12x1+4y130=9012x1+4y1=1203x1+y1=30(5)

Equation of line passing through origin, center and point (x1,y1) is

(y0)=(26)(x0)3y=x

Point (x1,y1) also lies on this line

3y1x1=0(6)

Solving (5),(6)

3x1+y1=30
9y13x1=0
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯10y1+0=30

y1=3,x1=9

The point (x1,y1) is (9,3)

892341_599185_ans_6c46262e6ee14379922e4debbd8d7c11.png

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