Position of a point relative to a circle
The equation of a circle can be used to check where a point lies relative to the circle.
!
Remember
A point can lie on, inside and outside a circle.
The distance from the point to the centre is then the same, smaller or larger than the radius.
The distance from the point to the centre is then the same, smaller or larger than the radius.
To determine the position of $P(x_0|y_0)$ from the centre $M(x_M|y_M)$, this point is inserted in the front part of the cartesian equation.
$(x_0-x_M)^2+(y_0-y_M)^2$
We now distinguish 3 cases. The result is
- $=r^2$: The point is on the circle.
- $<r^2$: The point is in the circle.
- $>r^2$: The point is outside the circle.
It is also easier to calculate with the magnitude of the vector $\vec{MP}$. (Attention, then it is checked for $r$, not $r^2$!)
- $|\vec{MP}|=r$: The point is on the circle.
- $|\vec{MP}|<r$: The point is in the circle.
- $|\vec{MP}|>r$: The point is outside the circle.
Example
Find the position of point $P(3|3)$ to circle $k$.
$k: (x-2)^2+(y+1)^2=25$
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Insert point
$(3-2)^2+(3+1)^2$ $=1^2+4^2$ $=17$
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Check
$17 \, ? \, r^2$
$17 < 25$
The point $P$ lies within the circle.