parabola-focus.js

const math = require('mathjs');
const LOOPS = 2.5 * 10 ** 7;
const xLim = 1;
const yLim = 1;
const r = Math.random;
let res = 0;

for (let i = 0; i < LOOPS; i++) {
  const p1 = { x: r(), y: r() };
  const p2 = { x: r(), y: r() };
  const p3 = { x: r(), y: r() };
  const slope_p1_p2 = (p2.y - p1.y) / (p2.x - p1.x);
  const slope_p2_p3 = (p3.y - p2.y) / (p3.x - p2.x);
  if (slope_p1_p2 != slope_p2_p3) {
    const b = [p1.y, p2.y, p3.y];
    const a = [
      [p1.x ** 2, p1.x, 1],
      [p2.x ** 2, p2.x, 1],
      [p3.x ** 2, p3.x, 1]
    ];
    try {
      const [A, B, C] = math.multiply(math.inv(a), b);
      const point = {};
      point.x = -B / (2 * A);
      point.y = (4 * A * C - B ** 2 + 1) / (4 * A);
      if (point.x >= 0 && point.x <= xLim && point.y >= 0 && point.y <= yLim)
        res++;
    } catch {}
  }
}

console.log(`Focus is in square: ${(res * 100) / LOOPS}%`);