Task list filter?
Choose required ranks and required tasks. The table of contents will list only tasks having one of the required ranks in corresponding rankings and at least one of the required tags (overall). If you wish to filter only according to some rankings or tags, leave the other groups empty.
Task rankings
Difficulty level
Task tags
Task type
«
«
Circle interpolation
Task number: 2355
Interpolate a circle through points A=(2,1), B=(4,3) and C=(0,7).
Hint
Use coordinates of points A, B and C in the circle equation (x−m)2+(y−n)2=r2. Subtract the quadratic terms to get the coordinates of the centre S=(m,n).
Resolution
We get equations
(2−m)2+(1−n)2=r2(4−m)2+(3−n)2=r2(0−m)2+(7−n)2=r2
After the expansion
−4m+m2−2n+n2+5=r2−8m+m2−6n+n2+25=r2m2−14n+n2+49=r2
When we subtract the third from the previous, we get a system of two linear equation with two unknowns:
−4m+12n−44=0−8m+8n−24=0
We first derive cordinates of the center S and then also the radius r.
Result
The circle equation is: (x−1)2+(y−4)2=10.
Observe that S is the center of the segment AC, hence the triangle ABC is right-angled.