randFromArray([[3, 4, 5], [6, 8, 10], [9, 12, 15], [5, 12, 13]]) randFromArray([[A, B], [B, A]]) 18 - X_AXIS 18 - Y_AXIS randRange(-MAX_X, MAX_X) randRange(-MAX_Y, MAX_Y) X_AXIS > Y_AXIS ? [H + C, K] : [H, K + C] X_AXIS > Y_AXIS ? [H - C, K] : [H, K - C] X_AXIS > Y_AXIS ? [[H, K - 1], [H, K + 1]] : [[H - 1, K], [H + 1, K]]

Find the foci of the ellipse below by moving the orange points to their correct positions.

Focus 1: `(` `FOCUS_START[0][0]``,\space` `FOCUS_START[0][1]``)`
Focus 2: `(` `FOCUS_START[1][0]``,\space` `FOCUS_START[1][1]``)`
Point to the ellipse to see the distance from that point to each focus.
Distance to focus 1 = `1`
Distance to focus 2 = `1`
Sum of distances = `1`
graph.f1 = F1; graph.f2 = F2; DUMMY_GRAPH = graph; initAutoscaledGraph([[-20, 20], [-20, 20]], {}); addMouseLayer(); graph.ellipse = interactiveEllipse({ center: [H, K], xRadius: X_AXIS, yRadius: Y_AXIS }); graph.focus1 = addMovablePoint({ coordX: FOCUS_START[0][0], coordY: FOCUS_START[0][1], snapX: 1, snapY: 1 }); graph.focus1.onMove = function(coordX, coordY) { var x = $("#problemarea span.focus1-x-label") $("#problemarea span.focus1-x-label").html("<code>" + coordX + "</code>").tex(); $("#problemarea span.focus1-y-label").html("<code>" + coordY + "</code>").tex(); }; graph.focus2 = addMovablePoint({ coordX: FOCUS_START[1][0], coordY: FOCUS_START[1][1], snapX: 1, snapY: 1 }); graph.focus2.onMove = function(coordX, coordY) { $("#problemarea span.focus2-x-label").html("<code>" + coordX + "</code>").tex(); $("#problemarea span.focus2-y-label").html("<code>" + coordY + "</code>").tex(); }; doEllipseInteraction(graph.ellipse, graph.focus1, graph.focus2); var writeDistances = function(coordX, coordY) { var focusDistance1 = KhanUtil.getDistance([coordX, coordY], graph.focus1.coord); var focusDistance2 = KhanUtil.getDistance([coordX, coordY], graph.focus2.coord); var distanceSum = focusDistance1 + focusDistance2; $("#problemarea span.focus-distance1").html("<code>" + round(10 * focusDistance1) / 10 + "</code>").tex(); $("#problemarea span.focus-distance2").html("<code>" + round(10 * focusDistance2) / 10 + "</code>").tex(); $("#problemarea span.distance-difference").html("<code>" + round(10 * distanceSum) / 10 + "</code>").tex(); }; graph.ellipse.writeDistances = writeDistances;
[DUMMY_GRAPH.focus1.coord, DUMMY_GRAPH.focus2.coord]
if (_.isEqual(guess, FOCUS_START)) { return "You need to move the foci to the correct positions."; } return (guess[0][0] === F1[0] && guess[0][1] === F1[1] && guess[1][0] === F2[0] && guess[1][1] === F2[1]) || (guess[0][0] === F2[0] && guess[0][1] === F2[1] && guess[1][0] === F1[0] && guess[1][1] === F1[1]);
graph.focus1.setCoord(guess[0]); graph.focus2.setCoord(guess[1]);

Both foci must lie along the major axis of the ellipse.

if (X_AXIS > Y_AXIS) { path([[H - X_AXIS, K], [H + X_AXIS, K]], { stroke: BLUE, strokeDasharray: "- " }); } else { path([[H, K - Y_AXIS], [H, K + Y_AXIS]], { stroke: BLUE, strokeDasharray: "- " }); }

The foci must be equally distant from the center of the ellipse.

circle([H, K], 0.25, { stroke: BLUE, fill: BLUE });

We can adjust the positions of the foci along the major axis until we've found the unique state where the sum of the distances is constant.

One focus is `(F1[0], F1[1])` and the other is `(F2[0], F2[1])`.

graph.focus1.moveTo(graph.f1[0], graph.f1[1]); graph.focus2.moveTo(graph.f2[0], graph.f2[1]);