Question

Let $f\left(x\right)=Ax+B,A,B\in R$ and $y=f\left(x\right)$ passes through the points $(A,2A-B^2)$ and $(2B+3,(A+B{)}^2-1)$. If $B_1,B_2\cdots B_n,n\in N,$ are different possible value(s) of $B$, then the value of $\sum _{r=1}^n{B}_r$ is

• A. $-\frac{9}{10}$
• B. $-\frac{18}{5}$
• C. $-\frac{9}{20}$
• D. $-\frac{9}{5}$

Answer 1

The correct option is D $-\frac{9}{5}$

Given : $f\left(x\right)=Ax+B$
Putting $(A,2A-B^2)$ and $(2B+3,(A+B{)}^2-1)$ in $y=f\left(x\right)$, we get
$2A-B^2=A^2+B\Rightarrow A^2+B^2-2A+B=0\cdots \left(1\right)(A+B{)}^2-1=2AB+3A+B\Rightarrow A^2+B^2-3A-B-1=0\cdots (2)$
Subracting equation $\left(2\right)$ from $\left(1\right)$, we get
$\Rightarrow A+2B+1=0\Rightarrow A=-(2B+1)$
Using equation $\left(1\right)$, we get
$\Rightarrow (2B+1{)}^2+B^2+2(2B+1)+B=0\Rightarrow 5B^2+9B+3=0$
There are $2$ possible values $B_1,B_2$, then
$\sum _{r=1}^2{B}_r=B_1+B_2=-\frac{9}{5}$