Problem: Find the equation of the circle with diameter AB, if A(-3,2) and B(1,4).
Problem: Find the equation of the circle with diameter AB, if A(-3,2) and B(1,4).
Solution:
Before moving to solution we should know that standard equation of a circle with radius ‘r’ and center at point O(a,b) is;
\[ (x-a)^2+(y-b)^2=r^2 \]
As it is given that AB is the diameter, so the mid point will be the center of the circle !
So, our first priority is to find the center of the circle !
Mid point of a straight line AB will be ;
\[ (\frac{-3+2}{2}, \frac{1+4}{2}) \]
or,
\[ (-1/2,5/2) \]
Now to find radius of the circle we proceed as below;
Diameter AB of the circle by distance formula will be,
\[ r=\sqrt{(2+3)^2+(4-1)^2} \]
\[ r=\sqrt{34} \]
Radius will be half of the diameter, so,
\[ r= \frac{\sqrt{34}}{2} \]
So the equation of circle will be,
\[ (x+\frac{1}{2})^2+(y-\frac{5}{2})^2=(\frac{\sqrt{34}}{2})^2 \]
After simplification we will get equation of the circle as;
\[ x^2+y^2+x-5y-2=0 \]
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