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Question: If 0 £ c £ 5, then the minimum distance of the point (0, c) from parabola y = x<sup>2</sup> is...

If 0 £ c £ 5, then the minimum distance of the point (0, c) from parabola y = x2 is

A

c–4\sqrt{c–4}

B

c–1/4\sqrt{c–1/4}

C

c+1/4\sqrt{c + 1/4}

D

None of these

Answer

c–1/4\sqrt{c–1/4}

Explanation

Solution

Point on parabola is (t\sqrt{t}, t)

distance = d2 = t + (t – c)2 = z.

z = t2 + t (1 – 2c) + c2

dzdt\frac{dz}{dt} = 2t + 1 – 2c

d2zdt2\frac{d^{2}z}{dt^{2}} = 2 > 0

if dzdt\frac{dz}{dt} = 0 Ž c – 12\frac{1}{2} = t

\ distance = (c–12)+(–12)2\sqrt{\left( c–\frac{1}{2} \right) + \left( –\frac{1}{2} \right)^{2}}= c–14\sqrt{c–\frac{1}{4}}