Multiple choice

If $a,b,c$ are in GP, then the equation $a{ x }^{ 2 }+2bx+c=0$ and $d{ x }^{ 2 }+2ex+f=0$ have a common root, if $\cfrac { d }{ a } ,\cfrac { e }{ b } ,\cfrac { f }{ c } $ are in

  1. AP

  2. GP

  3. HP

  4. A.G.P.
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A Correct answer
Explanation

If a, b, c are in GP, then b^2 = ac. The equation ax^2 + 2bx + c = 0 becomes ax^2 + 2bx + b^2/a = 0, which is (sqrt(a)x + b/sqrt(a))^2 = 0, so x = -b/a. Since this is a common root, it must satisfy the second equation. The condition for the coefficients leads to the conclusion that d/a, e/b, f/c are in AP.

AI explanation

Since a, b, c are in GP, we can substitute b^2 = ac, meaning the discriminant of ax^2 + 2bx + c = 0 is (2b)^2 - 4ac = 4b^2 - 4ac = 0, which reveals that both roots of this equation are equal to -b/a. For this equation to have a common root with dx^2 + 2ex + f = 0, the value -b/a must satisfy the second equation, leading to d(b^2/a^2) - 2e(b/a) + f = 0. Simplifying this and substituting b^2 = ac yields d(c/a) - 2e(b/a) + f = 0, or dc - 2eb + fa = 0. Dividing by abc gives d/b - 2e/c + f/a = 0, which rearranges to f/a + d/b = 2e/c, demonstrating that d/a, e/b, f/c are in Arithmetic Progression (AP).

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