Find the coordinates of the center of a circle with circles equations

Question

Let's say A(3;5) and B(-1;2) and Circle C with a diameter of [AB].
How can i find the coordinate of the center of the cricle ? I tried to make it with circles equation like:

(x-x')^2 + (y-y')^2 = r^2

but i got

y^2-16x+y = 1 

and so i don't know how to get the coordinates with that if this is right.

Answer 1

As you work your way through analytic geometry, don’t lose sight of the geometry part. Otherwise, you’ll find yourself floundering in a sea of algebra.

The midpoint of a diameter is the circle’s center. I expect that you know how to compute the midpoint of a line segment. I’m guessing that you’re looking for the coordinates of this point in order to plug it into the equation template $(x-x_c)^2+(y-y_c)^2=r^2$.

Note, though, that if you have the endpoints $A$ and $B$ of a diameter, you can write an equation of the circle down directly. The inscribed angle of a diameter is a right angle and the dot product of perpendicular vectors is equal to zero. Putting these two facts together we get the equation $$(x-x_a)(x-x_b)+(y-y_a)(y-y_b)=0$$ for the circle with diameter $AB$.

Answer 2

Center of circle as you mentioned in comments is the midpoint of line $AB$ : $ (h=1, k=3.5). $ This should come at first.

Next the length of diameter $AB=\sqrt{3^2+4^2} =5 = 2 r $ and radius $r=2.5$

and lastly the circle has equation

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