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.
(
-1,\space
1)
(
1,\space
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.
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.
(
-1,\space
1)
(
1,\space
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.