KVPY SB SX Exam 2011 Question Paper

Question : 83 of 120

Marks: +1, -0

The locus of the point P = (a, b) where a, b are real numbers such that the roots of

x^3 + ax^2 + bx + a = 0

are in arithmetic progression is

Solution:

Let roots

α-d, α, α + d

Sum:

3α=-a⇒α=-a/3

Product:

α(α^2-d^2)=-a

⇒α^2-d^2=3

pair product b

= α ^2 - αd + α ^2 + αd + α ^2 - d^2

b=2α^2+3

b-3=2/9a^2

⇒ locus

x^2=9/2(y-3)

parabola

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