Question

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

• A. Arithmetic Progession
• B. Geometric Progression
• C. Harmonic Progression
• D. GP or HP

Answer 1

The correct option is A

Arithmetic Progession

Solve it using Assumption

Put a=1,b=1,c=1.

Equation (1) is $x^2-2x+1=0$. Sum of the roots= 2 and product of roots= 1

Thus the roots are 1,1

Let the common root be 1

Let us assume the other root =2

The equation changes to

$x^2+3x+2=0$

Thus, $\frac{2e}{d}=3$ and $\frac{f}{d}=2$

When d=1, f=2 and $e=\frac{3}{2}$

Thus $\frac{d}{a}$; $\frac{e}{b}$; $\frac{f}{c}$ = 1, 1.5, 2 are in AP. Answer is option (a)