Download Rotated Ellipse The implicit equation x2

Differential Calculus
Grinshpan
Rotated Ellipse
The implicit equation
x2 − xy + y 2 = 3
describes an ellipse. The following 12 points are on this ellipse:
√
√
√ √
√
√
(± 3, 0), (0, ± 3), ( 3, 3), (− 3, − 3), (1, −1), (−1, 1), (1, 2), (2, 1), (−1, −2), (−2, −1).
The ellipse is symmetric about the lines y = x and y = −x.
It is inscribed into the square [−2, 2] × [−2, 2].
Solving the quadratic equation y 2 − xy + (x2 − 3) = 0 for y we obtain a pair of explicit
equations:
√
1
3 √
y= x ±
4 − x2 ,
2
2
where the plus corresponds to the top portion (red) and the minus corresponds to the bottom
portion (blue). Consequently, we obtain a formula for the slopes:
√
√
1
3
1
1
3
x
′
√
√
±
(−2x) =
∓
.
y =
2
2
2 2 4−x
2
2
4 − x2
For instance,
√
1
3
−1
′
√
y (−1,1) =
−
= 1.
2
2
4 − (−1)2
Alternatively, we can calculate the slopes using implicit differentiation:
2x − (y + xy ′ ) + 2yy ′ = 0
gives
y′ =
y − 2x
.
2y − x
1 − 2(−1)
Thus y ′ (−1,1) =
= 1. Note that the slopes at x = ±2 are infinite.
2 · 1 − (−1)