Assignment 6

10. Construct the locus of points equidistant from a fixed point F and a circle. In other words, repeat the parabola construction but use a circle as the "directrix." Let F be any point in the plane other than the center of the circle. Assume F is not on the circle; it can be either inside or outside.

First, explore this problem using below file. Drag the point F, then observe the locus of the point P.

If the point F is outside the given circle, what is the locus of the point P?  If the point F is inside the circle?

How to construct the locus?

Consider the case that a fixed point F is outside a given circle.

We should find a point (say P) equidistance from the point F and any point on the given circle. First, fix a point on the circle, and then move the point on the circle.

The point P should be on the line throughout the point A (the point that is on the given circle) and the center O of  the circle.

To find the point equidistance from A and F, we draw an isosceles triangle with AF as a base. Use the property that a perpendicular bisector of an isosceles passes throughout its vertex.

Like below figure, draw the line passing throughout A and O, and then draw the perpendicular bisector of the segment AF. Then, the intersection P of  two line is the third vertex of the isosceles triangle AFP.

The case that the point F is outside a given circle is similar.

As you observed, when the point F is outside the circle, the locus of P becomes a hyperbola, while F is inside the circle, the locus of P is an ellipse.

( If you want to explore this using GSP file, Click Here! )

Now, I prove why the locus of the point P becomes a hyperbola or ellipse.

<Case 1> The point F is outside a given circle.

I claim that

Since the triangle PFA is an isosceles, that is PA=PF,

(constant) ,

which means the definition of a hyperbola ( the difference of the distances from two fixed points is constant).

<Case 2> The point F is inside a given circle.

I claim that

Since the triangle PFA is an isosceles, that is PA=PF,

(Constant)

which means the definition of an ellipse ( the sum of the distances from two fixed points is constant).

Return to Main Page.