Question

If a, b, c are in G.P., then the equations \[a{x^2} + 2bx + c = 0\] and \[d{x^2} + 2ex + f = 0\] have a common root, if, \[\dfrac{d}{a},\dfrac{e}{b},\dfrac{f}{c}\] are in
1.A.P.
2.G.P.
3.H.P.
4.None of these

Answer 1

Hint: This question requires the basic knowledge about quadratic equations like the discriminant, common roots between 2 quadratic equations and the involvement of geometric progression in their roots. We need to know the formulae of discriminant and geometric progression in order to solve the problem:
 \[D = {b^2} - 4ac,\]
\[{b^2} = ac\]

Complete step-by-step answer:
Initially, we are given that a, b, c are in G.P. that means
\[ \Rightarrow {b^2} = ac - - - - (i)\]
Now, for the quadratic equation \[a{x^2} + 2bx + c = 0\] let’s find the discriminant \[D\]
\[ \Rightarrow D = {b^2} - 4ac\]
But on comparing the equation with the standard form of quadratic equation, we get, \[b = 2b\],
\[ \Rightarrow D = 4{b^2} - 4ac\]
Taking 4 common on the R.H.S we have
\[ \Rightarrow D = 4({b^2} - ac)\]
Using the equation \[(i)\] we get
\[ \Rightarrow D = 0\]
And when the discriminant of the quadratic equation is \[0\], Then both the roots of the equation are equal.
Now, the equations \[a{x^2} + 2bx + c = 0\]and \[d{x^2} + 2ex + f = 0\] have a common root which means both the roots of the equations are equal.
Thus, \[\dfrac{d}{a} = \dfrac{{2e}}{{2b}} = \dfrac{f}{c}\]
And so, if a, b, c are in G.P. then d, e, f will also be in the same progression.
Thus, d, e, f are in G.P.
Option (2) is the correct answer.
So, the correct answer is “Option 2”.

Note: This question requires a good use of the quadratic equation’s properties. One should be well versed with the concepts like progressions, to be able to solve this particular question. Take care of the calculations so as to be sure of the final answer. We must know the standard form of a quadratic equation.