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 equation of an ellipse is `\dfrac{(x - x_1)^2}{a^2} + \dfrac{(y - y_1)^2}{b^2} = 1`.

Find the foci of the ellipse below by moving the orange points to their correct positions. Then use that information to find the values of `x_1`, `y_1`, `a` and `b`.

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]);

`x_1 =` | H | `y_1 =` | K | `a =` | X_AXIS | `b =` | Y_AXIS

For all points on an ellipse, the sum of the distances to each focus is constant. There is only one way to arrange the two foci such that this is true.

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]);

`x_1` and `y_1` are the coordinates of the center of the ellipse.

The center of an ellipse is at the midpoint between its two foci.

`\left(\dfrac{F1[0] + F2[0]}{2}, \dfrac{F1[1] + F2[1]}{2}\right) = (H, K)`

So, `x_1 = H` and `y_1 = K`.

circle([H, K], 0.4, { stroke: null, fill: BLUE });

The `x`-axis of the ellipse is longer than its `y`-axis, so the sum of the distances to each focus is `2a`.

Therefore `a = X_AXIS`.

var x = 4.5; var scale = x / X_AXIS; var y = Y_AXIS * scale; diagram.scale = scale; init({ range: [[-5, 5], [-y - 0.05, y + 0.05]], scale: 25 }); ellipse([0, 0], [x, y], { stroke: BLUE }); var dx = C * scale; var dy = B * scale; path([[-C * scale, 0], [0, dy]], { stroke: GREEN }); path([[C * scale, 0], [0, dy]], { stroke: GREEN }); ellipse([dx, 0], [0.2, 0.2], { fill: ORANGE, stroke: null }); ellipse([-dx, 0], [0.2, 0.2], { fill: ORANGE, stroke: null }); label([-dx / 2 - 0.2, dy / 2 + 0.2], "a", {color: BLACK}); label([dx / 2 + 0.2, dy / 2 + 0.2], "a", {color: BLACK});

The `y`-axis of the ellipse is longer than its `x`-axis, so the sum of the distances to each focus is `2b`.

Therefore `b = Y_AXIS`.

var y = 4.5; var scale = y / Y_AXIS; var x = X_AXIS * scale; diagram.scale = scale; init({ range: [[-x - 0.05, x + 0.05], [-5, 5]], scale: 25 }); ellipse([0, 0], [x, y], { stroke: BLUE }); var dx = B * scale; var dy = C * scale; path([[0, -C * scale], [dx, 0]], { stroke: GREEN }); path([[0, C * scale], [dx, 0]], { stroke: GREEN }); ellipse([0, dy], [0.2, 0.2], { fill: ORANGE, stroke: null }); ellipse([0, -dy], [0.2, 0.2], { fill: ORANGE, stroke: null }); label([dx / 2, dy / 2 + 0.35], "b", {color: BLACK}); label([dx / 2, -dy / 2 - 0.35], "b", {color: BLACK});

The focal distance, `c` is the distance between the center of the ellipse to either focus.

Therefore `c = C`.

if (X_AXIS > Y_AXIS) { path([[0, 0], [diagram.scale * C, 0]], { stroke: BLACK }); label([diagram.scale * C / 2, -0.3], "c", {color: BLACK}); } else { path([[0, 0], [diagram.scale * B, 0]], { stroke: BLACK }); label([-0.3, diagram.scale * C / 2], "c", {color: BLACK}); }

`\qquad a^2 = b^2 + c^2`

label([-0.3, diagram.scale * B / 2], "b", {color: BLACK}); path([[0, 0], [0, diagram.scale * B]], { stroke: BLACK });
`\qquad\begin{align*} b^2 &= a^2 - c^2 \\ b &= \sqrt{a^2 - c^2} \\ b &= \sqrt{A^2 - C^2} \\ b &= \sqrt{A * A - C * C} \\ b &= B\end{align*}`

`\qquad a^2 = b^2 + c^2`

path([[0, 0], [0, diagram.scale * C]], { stroke: BLACK }); label([diagram.scale * B / 2, -0.35], "a", {color: BLACK});
`\qquad\begin{align*} a^2 &= b^2 - c^2 \\ a &= \sqrt{b^2 - c^2} \\ a &= \sqrt{A^2 - C^2} \\ a &= \sqrt{A * A - C * C} \\ a &= B\end{align*}`

So the equation of the ellipse is `\dfrac{(x - H)^2}{X_AXIS^2} + \dfrac{(y - K)^2}{Y_AXIS^2} = 1`.

`x_1 = H`, `y_1 = K`, `a = X_AXIS`, `b = Y_AXIS`.