Circle coordinate geometry question

circles

The question states:

Find the equation of the following circles: A circle has its centre on the line x + y = 1 and passes through the origin and the point (4,2).

What I did to try to solve this is draw it out and then I tried these two methods which were wrong

  1. Found a line with (0,0) and (4,2) and intersected it with x + y = 1. Then from there find the equation
  2. Tried and assumed that the line from (0,0) and (4,2) was the diameter and find the midpoint. Then from there find the equation.

So I was wondering a) why what I did was incorrect b) hints on how to solve it.

In general, I often struggle with a lot of circle coordinate geometry questions and was wondering if you had any tips on how to become better, or see it in a way where I would be able to answer it properly.

Thank you!

Best Answer

Dusting off your geometry book ...

Given any two points on a circle the center lies on to the perpendicular bisector of the chord between them. Here the perpendicular bisector passes through the midpoint $(1/2)((0,0)+(4,2))=(2,1)$. The slope of the chord is clearly $+1/2$ so, by the "negative reciprocal rule" the perpendicular line has slope $-2$. So the center lies on

$y-1=-2(x-2), y=-2x+5$

Since the center also lies on $y=1-x$ we then have for the center:

$1-x=-2x+5, x=4, y =-3$

meaning the center is $(4,-3)$. The radius should now be easy to figure out given that $(4,2)$ is on the circle centered at $(4,-3)$, and the equation of the circle follows.

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