# Question (Circles)

Consider S: x2 + y2 + (k-8)x + (2k-10)y + (1-4k) = 0, where k is a real number. As k varies, S represents a family of circles passing through two fixed points. Find the coordinates of these two points.

### 1 Answer

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- myisland8132Lv 71 decade agoFavorite Answer
S: x2 + y2 + (k-8)x + (2k-10)y + (1-4k) = 0

From the condition

Sub k=8

x2 + y2 + 6y = 31

Sub k=5

x2 + y2 -3x = 19

These two circle passing through two fixed points

So 31-6y=3x+19

3x+6y-12=0

x+2y-4=0

x=4-2y

Sub into x2 + y2 + 6y = 31

y^2-2y-3=0

y=-1 or 3

x=6 or -2

The coordinates of these two points are (6,-1) and (-2,3) respectively

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