Question

Assertion :Let equations $a{x}^2+bx+c=0(a,b,c\in R)$ & $x^2+2x+5=0$ have a common root, then $\frac{a+c}{b}=\frac{1}{3}$. Reason: If both roots of $A{x}^2+Bx+K_1=0$ & $A{}^{\prime }{x}^2+B{}^{\prime }x+K_2=0$ are identical, then $\frac{A}{A_1}=\frac{B}{B_1}=\frac{K_1}{K_2}$ (where $A,B,K_1$, and $A{}^{\prime },B{,}^{\prime }{K}_2\in R$).

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect, Reason is correct

Answer 1

The correct option is D Assertion is incorrect, Reason is correct

$f\left(x\right)=x^2+2x+5$
$D=4-20=-16$
Hence, it has imaginary roots.
Since it has real coefficients, the roots are conjugates.
Now, $a,b,c\in R$.
Since it has a common root with above equation, it too has imaginary roots, occurring in form of conjugates.
Thus, both the equations have equal roots.
Hence, $\frac{a}{1}=\frac{b}{2}=\frac{c}{5}=k$
Hence, $a=k$ $b=2k$ and $c=5k$.
Therefore,
$\frac{a+c}{b}$
$=\frac{6k}{2k}$
$=3$
Hence, assertion is incorrect.