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Mathematics
Let the parabola y=a x2+b x+c has vertex at M(4,2) and a ∈[1,3]. If the difference between the extreme values of abc is equal to N, then find the digit at unit place of N.
Q. Let the parabola
y
=
a
x
2
+
b
x
+
c
has vertex at
M
(
4
,
2
)
and
a
∈
[
1
,
3
]
. If the difference between the extreme values of abc is equal to
N
, then find the digit at unit place of
N
.
695
124
Conic Sections
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Answer:
6
Solution:
2
=
16
a
+
4
b
+
c
and
2
a
−
b
=
4
⇒
b
=
−
8
a
, hence,
c
=
16
a
+
2
∵
1
≤
a
≤
3
∴
18
≤
c
≤
50
when
a
=
1
,
b
=
−
8
, when
a
=
3
,
b
=
−
24
∴
N
m
a
x
=
18
×
1
×
−
8
N
m
i
n
=
50
×
3
×
−
24
∴
N
diff
=
3456