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Question

Consider a circle x2+y2+ax+by+c=0 lying completely in the first quadrant. If m1 and m2 are the maximum and the minimum values of yx for all ordered pairs (x,y) on the circumference of the circle, then the value of (m1+m2) is

A
2ab4cb2
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B
2abb24ac
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C
2abb24c
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D
2ab4acb2
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Solution

The correct option is A 2ab4cb2
Given x2+y2+ax+by+c=0
Let m=yxy=mx
Now, here m=yx represents slope of the line passing through O(0,0) and a point on the circle.

Substituting y=mx inx2+y2+ax+by+c=0, we get
x2+m2x2+ax+bmx+c=0(1+m2)x2+(a+bm)x+c=0

Applying discriminant =0 ( For maximum and minimum value of m, line y=mx should be tangent to the circle)
(a+bm)24(1+m2)c=0
m2(b24c)+2abm+a24c=0

So, we get two values of m, i.e. m1 and m2, for maximum and minimum.
m1+m2=2abb24c=2ab4cb2

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