# "All-in-one" loci: ellipse, hyperbola and circle

- Author:
- Simona Riva

Move the foci

*F*_{1}and*F*_{2}that define the position of the focal axis of the conic, set the length of the*major axis*of the conic using the*Maxis*slider, then move point*P*along the circle to view the locus. To reproduce the construction:- Draw the circle with center
*F*_{1}and radius*r*=*Maxis* - Create a point
*P*on the circle, then draw the perpendicular bisector of segment*P**F*_{2} - Line
*P**F*_{1 }intersects the perpendicular bisector at a point*L*, which is a point of the conic

*P*, moving on the circle, creates the locus of points*L*, that is: - an*ellipse*if*F*_{2}is*inside*the circle*distance between foci*<*axis length*→*e*< 1 - an*hyperbola*if*F*_{2 }is*outside*the circle* distance between foci*>*axis length*→*e*> 1 - a*circle*if*F*_{1}≡*F*_{2} *distance between foci*= 0 →*e*= 0 (*e*=*eccentricity*)## Explore the locus

After creating the locus, draw triangle

*PLF*._{2}- What type of triangle is it? Explain.
- Write the canonical definition (as locus) of the conic that you see in the construction.
- Use the properties of triangle
*PLF*to show that the graph in the construction matches the canonical definition._{2}