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Nature of Roots

If roots of t...

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If roots of the quadratic equation a(b-c)x^2+b(c-a)x+c(a-b)=0 are equal and a,b,c is greater than 2÷b=1÷a+1÷c ie a,b,c are in h.p

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Q. If the roots of the equation

a (b - c) x^{2} + b (c - a) x + c (a - b) = 0

are equal then prove that

\frac{2}{b} = \frac{1}{a} + \frac{1}{c}

a, b, c

a, b, c

are rational and no two of them are equal, then the quadratic equations

(b - c) x^{2} + (c - a) x + a - b = 0

a (b - c) x^{2} + b (c - a) x + c (a - b) = 0

a, b, c

are in H.P., then the equation

a (b - c) x^{2} + b (c - a) x + c (a - b) = 0

Q. If roots of quadratic equation

(b - c) x^{2} + (c - a) x + (a - b) = 0

are real and equal then prove that

2 b = a + c

Q. (a) If the roots of the equation,

(b - c) x^{2} + (c - a) x + (a - b) = 0

be equal, then prove thar a,b,c are in arithmetical progression.
(b) If

a (b - c) x^{2} + b (c - a) x + c (a - b) = 0

has equal roots, prove that a,b,c are in harmonical progression.

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