Answer:
Explanation:
The tangent line is parallel to the
x
axis when the slope (hence
d
y
d
x
) is zero and it is parallel to the
y
axis when the slope (again,
d
y
d
x
) goes to
∞
or
−
∞
Explanation:
We'll start by finding
d
y
d
x
:
x
2
+
x
y
+
y
2
=
7
d
d
x
(
x
2
+
x
y
+
y
2
)
=
d
d
x
(
7
)
2
x
+
1
y
+
x
d
y
d
x
+
2
y
d
y
d
x
=
0
d
y
d
x
=
−
2
x
+
y
x
+
2
y
Now,
d
y
d
x
=
0
when the nuimerator is
0
, provided that this does not also make the denominator
0
.
2
x
+
y
=
0
when
y
=
−
2
x
We have now, two equations:
x
2
+
x
y
+
y
2
=
7
y
=
−
2
x
Solve (by substitution)
x
2
+
x
(
−
2
x
)
+
(
−
2
x
)
2
=
7
x
2
−
2
x
2
+
4
x
2
=
7
3
x
2
=
7
x
=
±
√
7
3
=
±
√
21
3
Using
y
=
−
2
x
, we get
The tangent to the curve is horizontal at the two points:
(
√
21
3
,
−
2
√
21
3
)
and
(
−
√
21
3
,
2
√
21
3
)
(Observe that these pair do not also make the denominator of
d
y
d
x
equal to
0
)
To find the points at which the tangent is vertical, make the denominator of
d
y
d
x
equal tpo
0
(without also making the numerator
0
).
We could go through the solution, but the symmetry of the equation that we will get:
x
=
−
2
y
, so
y
=
±
√
21
3
and the points on the curve at which the tangent is vertical are:
(
−
2
√
21
3
,
√
21
3
)
and
(
2
√
21
3
,
−
√
21
3
)
By the way. Because we do have the technology, here is the graph of this rotated ellipse: (Note that
±
√
21
3
≈
±
1.528
which you can see on the graph.)
graph{x^2 + xy +y^2 =7 [-11.3, 11.2, -5.665, 5.585]}
Answer link