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]]
The sum of the distances between each point on an ellipse and the ellipse's two foci is always the same.
Find the foci of the ellipse below by moving the orange points to their correct positions.
You can check your answer by moving your cursor across the ellipse's perimeter and seeing how the sum of distances from the foci changes.
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]);