327
Answers
501–600
Answers and Explanations
Using x = − 1
7
and y
2
= 8 – 8x
2
, find the y coordinate:
y
y
y
y
2
2
2
2
8 8 1
7
8 8
49
384
49
384
49
8 6
7
= − −
= −
=
= ±
= ±
( )
Therefore, the two points that are farthest from (1, 0) are the points −1
7
8 6
7
,

 

  and
− −
1
7
8 6
7
,

 

  .
505.
−24
17
6
17
,
(
)
You want to find the minimum distance. The distance from a point (x, y) to the origin is
D x y
x
y
x
y
( , )
(
)
=
−
+ ( − )
=
+
0
0
2
2
2
2
Using y = 4x + 6, the distance is
D x
x
x
x
x
( ) =
+( + )
=
+
+
2
2
2
4
6
17
48
36
Tip: You can take the derivative of this function and use the first derivative test
to find a minimum, but it’s easier to use the square of the distance, which gets rid of
the radical. For a function that satisfies f
 
(x) ≥ 0, its local maxima and minima occur at
the same x values as the local maxima and minima of its square, [f
 
(x)]
2
. Obviously,
the corresponding y values would change, but that doesn’t matter here!
Using the square of the distance, you have
S
x
x
=
+
+
17
48
36
2
The derivative is
′
S
x
=
+
34
48
Setting the derivative equal to zero gives you 34x + 48 = 0, which has the solution x = −24
17
.
You can verify that this value gives you a minimum by using the first derivative test.
Using x = −24
17
and y = 4x + 6, find the y coordinate:
y = −
+
= − +
=
4 24
17
6
96
17
102
17
6
17
( )
Therefore, the point closest to the origin is −24
17
6
17
,
(
) .