Question

If $a,b,c$ be the $4^{th},7^{th}and{10}^{th}$ term of an AP respectively then the sum of the roots of the equation $a{x}^2-2bx+c=0$

• A. $-\frac{b}{a}$
• B. $-\frac{2b}{a}$
• C. $\frac{c+a}{a}$
• D. None of these

Answer 1

The correct option is C

$\frac{c+a}{a}$

The explanation for the correct option (Option C):

Let $xandd$ be the first term and the common difference of the AP.

The general term is given by $a_n=x+\left(n-1\right)d$.

Step 1: Defining values of $a,b,c$

$$\begin{gathered} a_4=x+3d=a...\left(1\right) \\ {a}_7=x+6d=b...\left(2\right) \\ {a}_{10}=x+9d=c...\left(3\right) \end{gathered}$$

Step 2: Equating (1) and (2),

$$\begin{gathered} 6d-3d=b-a \\ \Rightarrow d=\frac{b-a}{3} \end{gathered}$$

Substituting in (1), we get

$x=2a-b$

From equation (3), we get

$$\begin{gathered} x+9d=c \\ \Rightarrow \left(2a-b\right)+9\left(\frac{b-a}{3}\right)=c \\ \Rightarrow 2a-b+3b-3a=c \\ \Rightarrow -a+2b=c \\ \Rightarrow 2b=a+c \end{gathered}$$

Step 3: Sum of the roots of the equation $a{x}^2-2bx+c=0$:

$$\begin{gathered} \mathrm{Sum}\mathrm{of}\mathrm{roots}ofequationa{x}^2+bx+c=0is-\frac{b}{a} \\ Therefore,\mathrm{Sum}\mathrm{of}\mathrm{roots}=\frac{2b}{a}=\frac{a+c}{a} \end{gathered}$$

Hence, the correct answer is option C.