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

In summary, to find the equation of a circle that passes through points A(2,2) and B(5,3) and has its center on the line y=x+1, we can use the equation (x-h)^2 + (y-k)^2 = r^2 and plug in the coordinates of the two points to get two equations with three variables. To solve for these variables, we can use the fact that any perpendicular bisector of a chord is a radius, meaning it passes through the center of the circle. We can find the center point of the interval AB by finding the midpoint of the line segment AB, and the slope of the line AB can be used to find the slope of a line perpendicular to it
  • #1
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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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  • #2
What is h in terms of k?
 
  • #3
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?
 
  • #4
thanks!
 

What is the formula for finding the center of a circle passing through two points?

The formula for finding the center of a circle passing through two points is (x,y), where x is the average of the x-coordinates of the two given points and y is the average of the y-coordinates of the two given points.

How do you determine the radius of the circle passing through two points?

The radius of the circle can be found by calculating the distance between the two given points, which serves as the diameter of the circle. The radius is then half the length of the diameter.

Can the center of the circle lie on any line passing through the two given points?

No, the center of the circle must lie on the perpendicular bisector of the line segment connecting the two given points. This ensures that the center is equidistant from the two points, fulfilling the definition of a circle.

What is the significance of finding a circle passing through two points with its center lying on a line?

Finding such a circle can be useful in various applications, such as in geometry and engineering. It allows for the creation of circular shapes that are tangent to a given line, which can be helpful in construction and design.

Are there any special cases when finding a circle passing through two points with its center lying on a line?

Yes, there are two special cases to consider. First, if the two given points are the same, then the circle is simply a point. Second, if the two given points are equidistant from the given line, then the center of the circle lies on the given line itself, and the radius is zero.

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