## Fancies of the parabola and hyperbola

Many revolutions-of-the-sun ago we were seeking a device that could construct a parabola. We made a discovery in this connection that is exceedingly elementary Euclid for mathematicians, but for us it was an insight-giving philosophical revelation. A parabola can be made thus:1) Draw a line L1;

2) Take a point P on it;

3) Draw perpendicular line P1 through P;

4) Take a random fixed point F, which will be the focus of the future parabola;

5) Draw the segment FP;

6) Draw the perpendicular bisector line P2 of segment FP;

7) Mark the point Q where P2 intersects P1;

8) Track the locus of Q as P moves on P1, i.e. the locus of the intersection of the perpendicular bisector P2 of FP with P1 as P moves on L1;

This locus is a parabola with focus at F. All the lines P2 are tangents of this parabola.

We can obtain the equation of the parabola thus:

Place origin O (0,0) at the minimum of the parabola;

Then the focus F is at (0,a);

Because P2 is the perpendicular bisector of FP we can show that the equation of line L1 is y=-a using similarity of triangles;

Coordinates of the point on the generated locus are Q(x,y);

Hence, P(x,-a);

By baudhAyana’s relationship: FQ^2=PQ^2; hence, (x-0)^2+(y-a)^2=(x-x)^2+(y+a)^2;

Thus we have y=x^2/(4*a) as the equation of the parabola (or proof that the locus of Q is a parabola).

We then found that if you repeat the above procedure with a circle C1 in place of line L1, and a radial line P1 in place of the P1 of the above example we get a hyperbola: