Find circle passing through two points and center lying on a line

AI Thread Summary
To find the equation of a circle that passes through points A(2,2) and B(5,3) with its center on the line y = x + 1, two equations can be derived from the circle's standard form using the coordinates of points A and B. This results in two equations involving the center coordinates (h, k) and the radius r. To solve for the center, one can determine the midpoint of segment AB and the slope of line AB, then find the slope of the perpendicular bisector. The intersection of this bisector with the line y = x + 1 will provide the center of the circle. The discussion emphasizes using geometric properties and relationships to solve for the circle's parameters.
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Homework Statement



Find the equation of a circle that passes through the points A(2,2) and B(5,3) and has its centre on the line y = x +1

Homework Equations



(x-h)^2 + (y-k)^2 = r^2

The Attempt at a Solution



can get 2 equations knowing the 2 points the circle passes through but still have 3 variables and am not sure how to use the equation for the centre

(2-h)^2 + (2-k)^2 = r^2

(5-h)^2 + (3-k)^2 = r^2


How do I solve from here?
 
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What is h in terms of k?
 
Any perpendicular bisector of a chord is a radius- i.e. passes through the center of the circle.

What is the center point of the interval AB? What is the slope of the line AB? What is the slope of a line perpendicular to that? What is the equation of the perpendicular bisector of AB? Where does that line intersect y= x+1?
 
thanks!
 

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