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Byju's Answer
Standard XII
Mathematics
Range of Quadratic Expression
Find the poin...
Question
Find the point on the curve x
2
=8y which is nearest to the point (2,4).
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Solution
Let
x
,
y
be
nearest
to
the
point
2
,
4
.
Then
,
x
2
=
8
y
⇒
y
=
x
2
8
.
.
.
1
d
2
=
x
-
2
2
+
y
-
4
2
Using
distance
formula
Now
,
Z
=
d
2
=
x
-
2
2
+
y
-
4
2
⇒
Z
=
x
-
2
2
+
x
2
8
-
4
2
From
eq
.
1
⇒
Z
=
x
2
+
4
-
4
x
+
x
4
64
+
16
-
x
2
⇒
d
Z
d
y
=
-
4
+
4
x
3
64
For
maximum
or
minimum
values
of
Z
,
we
must
have
d
Z
d
y
=
0
⇒
-
4
+
4
x
3
64
=
0
⇒
x
3
16
=
4
⇒
x
3
=
64
⇒
x
=
4
Substituting
the
value
of
x
in
eq
.
1
,
we
get
y
=
2
Now
,
d
2
Z
d
y
2
=
12
x
2
64
⇒
d
2
Z
d
y
2
=
3
>
0
So
,
the
nearest
point
is
4
,
2
.
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0
Similar questions
Q.
Determine the points on the curve x2 = 4y which are nearest to the point (0,5).
Q.
The point on the curve
x
2
= 2
y
which is nearest to the point (0, 5) is
(A)
(B)
(C) (0, 0) (D) (2, 2)