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If the sum of the roots of the quadratic equation ax2+bx+c = 0, is equal to the sum of the squares of their reciprocals, then (a/c), (b/a), (c/b) are in
  • a)
    arithmetic progression
  • b)
    geometric progression
  • c)
    harmonic progression
  • d)
    arithmetico-geometric progression
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If the sum of the roots of the quadratic equation ax2+bx+c = 0, is equ...
The quadratic equation ax^2 + bx + c = 0 has roots x1 and x2. The sum of the roots is given by x1 + x2. The sum of the squares of the reciprocals of the roots is given by 1/x1^2 + 1/x2^2.
If the sum of the roots is equal to the sum of the squares of their reciprocals, we can set up the following equation:

x1 + x2 = 1/x1^2 + 1/x2^2

This equation can be rearranged as follows:

x1^2 + x2^2 = x1 + x2

We can rewrite this equation as:

(x1 + x2)^2 - 2x1x2 = x1 + x2

This equation can be rearranged as:

(x1 + x2)^2 - (x1 + x2) - 2x1x2 = 0

This is a quadratic equation in the form ax^2 + bx + c = 0, where a = -2x1x2, b = -(x1 + x2), and c = (x1 + x2)^2.

If the sum of the roots of a quadratic equation is equal to the sum of the squares of their reciprocals, then the coefficients a, b, and c of the quadratic equation will be in a harmonic progression.

Therefore, (a/c), (b/a), (c/b) are in:

c) harmonic progression

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If the sum of the roots of the quadratic equation ax2+bx+c = 0, is equal to the sum of the squares of their reciprocals, then (a/c), (b/a), (c/b) are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)arithmetico-geometric progressionCorrect answer is option 'C'. Can you explain this answer?
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