Create assignment, 57950, Exam 2, Oct 25 at 1:12 pm
2
The points
P
and
Q
on the graph of
y
2
−
xy
+ 6 = 0
have the same
x
-coordinate
x
= 5. Find the
point of intersection of the tangents to the
graph at
P
and
Q
.
1.
intersect at =
12
5
,
24
5
2.
intersect at =
12
5
,
12
5
3.
intersect at =
24
5
,
12
5
correct
4.
intersect at =
24
5
,
24
5
5.
intersect at =
12
5
,
4
5
Explanation:
The
y
-coordinate at
P, Q
will be the solu-
tions of
(
‡
)
y
2
−
xy
+ 6 = 0
at
x
= 5,
i.e.
, the solutions of
y
2
−
5
y
+ 6 = (
y
−
3)(
y
−
2) = 0
.
Thus
P
= (5
,
3)
,
Q
= (5
,
2)
.
To determine the tangent lines we need also
the value of the derivative at
P
and
Q
. But
by implicit differentiation,
2
y
dy
dx
−
x
dy
dx
−
y
= 0
.
so
dy
dx
=
y
2
y
−
x
.
Thus
dy
dx
P
= 3
,
dy
dx
Q
=
−
2
.
By the point-slope formula, therefore, the
equation of the tangent line at
P
is
y
−
3 = 3(
x
−
5)
,
while that at
Q
is
y
−
2 =
−
2(
x
−
5)
.
Consequently, the tangent lines at
P
and
Q
are
y
−
3
x
=
−
12
and
y
+ 2
x
= 12
respectively. These two tangent lines
intersect at =
24
5
,
12
5
.
StewartC5 03 07 37
49:05, calculus3, multiple choice,
>
1 min,
wording-variable.
004
Find an equation of the tangent line to the
ellipse
x
2
a
2
+
y
2
b
2
= 1
at the point (
x
o
, y
o
).
1.
x
o
x
a
2
+
y
o
y
b
2
= 1
correct
2.
x
o
x
a
2
−
y
o
y
b
2
=
−
1
3.
x
a
2
−
y
b
2
=
x
o
y
o
4.
x
a
2
+
y
b
2
=
−
1
5.
x
o
x
a
2
+
y
o
y
b
2
=
x
o
y
o
Explanation:
x
2
a
2
+
y
2
b
2
= 1
2
x
a
2
+
2
yy
b
2
= 0
y
=
−
b
2
x
a
2
y