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Question

The point on parabola 2y=x2, which is nearest to the point (0,5) is

A
(4,8)
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B
(1,1/2)
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C
(22,4)
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D
None of these
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Solution

The correct option is C (22,4)
Let (h,k) lie on the curve x2=2y which is nearest to the point (0,5)
Since (h,k) lie on the curve x2=2y
(h,k) will satisfy the equation of curve x2=2y
Putting x=h and y=k in equation we get
h2=2k ......(1)
We need to minimize the distance of a point (h,k) from (0,5)
Let D be the distance between (h,k) and (0,5)
D=h2+(5k)2
From (1) we have h2=2k
D=2k+(5k)2
Differentiating w.r.t k we get
dDdk=122k+(5k)2×ddk(2k+(5k)2)
=122k+(5k)2×[2+2(5k)ddk(5k)]
=122k+(5k)2×[22(5k)]
=4+k2k+(5k)2
Put dDdk=0
4+k2k+(5k)2=0
4+k=0
k=4 is a point of minima.
D is minimum when k=4
We have h2=2k=2×4=8
h=22
Hence the required point is (h,k)=(22,4)

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