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

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Homework Help Overview

The problem involves finding the equation of a circle that passes through two given points, A(2,2) and B(5,3), with the additional constraint that the center of the circle lies on the line defined by y = x + 1.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to derive equations based on the two points the circle passes through but notes the challenge of having three variables with only two equations. A participant questions the relationship between the variables h and k. Another participant suggests considering the properties of the perpendicular bisector of the chord formed by points A and B, prompting further exploration of geometric relationships.

Discussion Status

The discussion is active, with participants exploring different geometric properties and relationships relevant to the problem. Guidance has been offered regarding the use of the perpendicular bisector and its intersection with the line y = x + 1, although no consensus or resolution has been reached yet.

Contextual Notes

The original poster expresses uncertainty about how to proceed with the problem given the constraints and the number of variables involved.

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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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