randFromArray([[3, 4, 5], [6, 8, 10], [9, 12, 15], [5, 12, 13]]) A * A B * B 18 - C randRange(-5, 5) randRange(-MAX_Y, MAX_Y)

The equation of a north-south opening hyperbola is :
`\qquad \dfrac{(x - x_1)^2}{b^2} - \dfrac{(y - y_1)^2}{a^2} = 1`.

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

Focus 1: `(` `-1``,\space` `1``)`
Focus 2: `(` `1``,\space` `1``)`
Point to the hyperbola to see the distance from that point to each focus.
Distance to focus 1 = `1`
Distance to focus 2 = `1`
Difference of distances = `1`
graph.H = H; graph.K = K; graph.C = C; initAutoscaledGraph([[-20, 20], [-20, 20]], {}); addMouseLayer(); graph.focus1 = addMovablePoint({ coordX: -2, coordY: 2, snapX: 1, snapY: 1 }); graph.focus1.onMove = function(coordX, coordY) { $("#problemarea span.focus1-x-label").html("<code>" + coordX + "</code>").tex(); $("#problemarea span.focus1-y-label").html("<code>" + coordY + "</code>").tex(); }; graph.focus2 = addMovablePoint({ coordX: 2, coordY: 2, 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(); }; graph.func1 = addInteractiveFn(function(x) { return K + sqrt(A2 / B2 * (x - H) * (x - H) + A2); }, {}); graph.func2 = addInteractiveFn(function(x) { return K - sqrt(A2 / B2 * (x - H) * (x - H) + A2); }, {}); doHyperbolaInteraction(graph.func1, graph.focus1, graph.focus2); doHyperbolaInteraction(graph.func2, 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 difference = abs(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 * difference) / 10 + "</code>").tex(); }; graph.func1.writeDistances = writeDistances; graph.func2.writeDistances = writeDistances;
[graph.focus1.coord, graph.focus2.coord]
if (_.isEqual(guess, [[-2, 2], [2, 2]])) { return ""; } return guess[0][0] === H && guess[1][0] === H && (guess[0][1] === K - C && guess[1][1] === K + C) || (guess[0][1] === K + C && guess[1][1] === K - C)
graph.focus1.setCoord(guess[0]); graph.focus2.setCoord(guess[1]);

Equation of the hyperbola:

`y_1 =` | K
`x_1 =` | H
`a =` | A
`b =` | B

For all points on a hyperbola, the absolute value of the distance from one focus minus the distance to the other focus is equal. There is only one way to arrange the two foci such that this is true.

One focus is `(H, K + C)` and the other is `(H, K - C)`.

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

The center of a hyperbola is at the midpoint of its two foci.

`\left(\dfrac{H + H}{2}, \dfrac{K+C + K-C}{2}\right) = (H, K)`

The absolute value of the distance from one focus minus the distance to the other focus is equal to `2a`

`\qquad\begin{align*} 2a &= 2 * A \\ a &= A\end{align*}`

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

`\qquad\begin{align*} c^2 &= a^2 + b^2 \\ b^2 &= c^2 - a^2 \\ b &= \sqrt{c^2 - a^2} \\ b &= \sqrt{C^2 - A^2} \\ b &= \sqrt{C * C - A * A} \\ b &= B\end{align*}`

So the equation of the hyperbola is `\dfrac{(y - K)^2}{A^2} - \dfrac{(x - H)^2}{B^2} = 1`.

randFromArray([[3, 4, 5], [6, 8, 10], [9, 12, 15], [5, 12, 13]]) A * A B * B 18 - C randRange(-5, 5) randRange(-MAX_Y, MAX_Y)

The equation of an east-west opening hyperbola is:
`\qquad \dfrac{(x - x_1)^2}{b^2} - \dfrac{(y - y_1)^2}{a^2} = 1`.

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

Focus 1: `(` `-1``,\space` `1``)`
Focus 2: `(` `1``,\space` `1``)`
Point to the hyperbola to see the distance from that point to each focus.
Distance to focus 1 = `1`
Distance to focus 2 = `1`
Difference of distances, `c` = `1`
graph.H = H; graph.K = K; graph.C = C; initAutoscaledGraph([[-20, 20], [-20, 20]], {}); addMouseLayer(); graph.focus1 = addMovablePoint({ coordX: -2, coordY: 2, snapX: 1, snapY: 1 }); graph.focus1.onMove = function(coordX, coordY) { $("#problemarea span.focus1-x-label").html("<code>" + coordX + "</code>").tex(); $("#problemarea span.focus1-y-label").html("<code>" + coordY + "</code>").tex(); }; graph.focus2 = addMovablePoint({ coordX: 2, coordY: 2, 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(); }; graph.func1 = addInteractiveFn(function(x) { return K + sqrt(A2 / B2 * (x - H) * (x - H) + A2); }, { swapAxes: true }); graph.func2 = addInteractiveFn(function(x) { return K - sqrt(A2 / B2 * (x - H) * (x - H) + A2); }, { swapAxes: true }); doHyperbolaInteraction(graph.func1, graph.focus1, graph.focus2); doHyperbolaInteraction(graph.func2, 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 difference = abs(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 * difference) / 10 + "</code>").tex(); }; graph.func1.writeDistances = writeDistances; graph.func2.writeDistances = writeDistances;
[graph.focus1.coord, graph.focus2.coord]
if (_.isEqual(guess, [[2, -2], [2, 2]])) { return ""; } return guess[0][1] === H && guess[1][1] === H && (guess[0][0] === K - C && guess[1][0] === K + C) || (guess[0][0] === K + C && guess[1][0] === K - C)
graph.focus1.setCoord(guess[0]); graph.focus2.setCoord(guess[1]);

Equation of the hyperbola:

`x_1 =` | K
`y_1 =` | H
`b =` | B
`a =` | A

For all points on a hyperbola, the absolute value of the distance from one focus minus the distance to the other focus is equal. There is only one way to arrange the two foci such that this is true.

One focus is `(K - C, H)` and the other is `(K + C, H)`.

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

The center of a hyperbola is at the midpoint of its two foci.

`\left(\dfrac{K - C + K + C}{2}, \dfrac{H + H}{2}\right) = (K, H)`

The absolute value of the distance from one focus minus the distance to the other focus is equal to `2a`

`\qquad\begin{align*} 2a &= 2 * A \\ a &= A\end{align*}`

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

`\qquad\begin{align*} c^2 &= a^2 + b^2 \\ b^2 &= c^2 - a^2 \\ b &= \sqrt{c^2 - a^2} \\ b &= \sqrt{C^2 - A^2} \\ b &= \sqrt{C * C - A * A} \\ b &= B\end{align*}`

So the equation of the hyperbola is `\dfrac{(x - H)^2}{B^2} + \dfrac{(y - K)^2}{A^2} = 1`.