Circles

Point on Circle

Test whether a point satisfies a circle equation by substitution.

point-on-circle A point is on the circle when substituting its coordinates makes the left side equal r squared.

Example

Substitute a point and compare with the circle radius.

highlighted = computed this step

Step 1 — Plot point and circle

Plot the point and circle.

Coordinate plot\text{Coordinate plot}
Point on circle plotPoint P(5, 1) compared with circle centered at C(2, -3).CP

Step 2 — Circle equation

Read the circle equation.

(x−2)2+(y+3)2=25(x- \hl{2} )^{2} + (y+ \hl{3} )^{2} = \hl{25}

Step 3 — Test point

Read the test point.

P=(5,1)P=( \hl{5} , \hl{1} )

Step 4 — Substitute point

Substitute the point into the circle.

(5−2)2+(1+3)2=25( 5 - 2 )^{2} + ( 1 + 3 )^{2} = \hl{25}

Step 5 — Verdict

Compare with the radius squared.

25=25⇒on circle25 = 25 \Rightarrow\text{on circle}