Parabola. Propositions and instructions. Propositions 12-13.

The canonical equation of the parabola is y² = 2*p*x. Proposition 12. The distance from an arbitrary point M(x,y) on the parabola to the focus is equal to r=x+p/2. Note. Actually the square of distance from M to F is equal to r2=(x-p/2)2+y2=(x-p/2)2+2*p*x=(x+p/2)2. Hence, due to the fact that x>=0, it follows that r=x+p/2. Proposition 13. The parabola is the locus of points equally distant from the focus and the directrix.