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Vivek @ Born this Way
·2012-03-19 19:59:51
2. \textbf{The minimum No. of points required to uniquely determine a parabola is}
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Aditya Bhutra
·2012-03-20 00:30:49
2) i think it is 4
1) it is easy to find that f(2)<0
thus 212 -2ab +a2 <0
or 212 +(a-b)2 <b2
for minimum value of |b| , a=b
thus b2>212
or |b| >26 =64
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Vivek @ Born this Way
·2012-03-20 00:48:14
But why f(2) < 0 ,and what if I say a ≠b ?
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Swaraj Dalmia
·2012-03-20 01:07:50
2)ans =4
general equation is (y-k)2=4a(x-h)
to solve this we need 3 pts.[3 variables]
But this will give 2 values of k.
We need a 4th to find 1 value of k.
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Aditya Bhutra
·2012-03-20 01:25:22
@vivek - the given equation can have a maximum of two real roots. and as the leading coeff. is +ve , thus it will almost be like an upward facing curve (like a parabola) .
now since 2 lies between the roots, f(2) <0
and if a≠b then min |b| will be slightly greater (but its limiting value will still remain the same)
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Shubhodip
·2012-03-20 03:22:55
2) It is three and not four
It is a second degree polynomial, so enough to know it at three points
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Hari Shankar
·2012-03-20 04:49:59
1) The problem should read,"if for some real value of a, one root of the equation ..."
Lets assume the converse i.e. that the given polynomial has all roots less than 2 (or greater than 2) for every real a
Since its of even degree with positive leading coefficient, this means we must have f(2)>0
i.e. 2^{12}-2ab+a^2>0
This is independent of the choice of real a. Viewing it as a quadratic in a, this means we must have determinant negative.
i.e. |b|<26.
This means that if b≥26, for some choice of a, f(2)<0 i.e. there exists an odd number of roots of f greater than 2 for some real values of a.