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_START[0][0]
,\space
FOCUS_START[0][1]
)
(
FOCUS_START[1][0]
,\space
FOCUS_START[1][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])
.
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
.
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
.
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
.
The focal distance, c
is the distance between the center of the ellipse to either focus.
Therefore c = C
.
\qquad a^2 = b^2 + c^2
\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
\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
.