Question

Assertion :If $a,b,c\in R$ and equations $a{x}^2+bx+c=0$ & $x^2+3x+4$ have a common root then $\frac{a+c}{b}=\frac{4}{3}$. Reason: If both roots of $a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$ and $a^{\prime \prime }{x}^2+b^{\prime \prime }x+c^{\prime \prime }=0$ are identical then $\frac{a^{\prime }}{a^{\prime \prime }}=\frac{b^{\prime }}{b^{\prime \prime }}=\frac{c^{\prime }}{c^{\prime \prime }}$ where $a{,}^{\prime }b{,}^{\prime }c{,}^{\prime \prime }b{,}^{\prime \prime }c,\in R$.

• 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 but Reason is correct

Answer 1

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

Given equations, $a{x}^2+bx+c=0$
$x^2+3x+4=0$Here, $D=9-16=-7<0$ , so complex roots.
Since, roots of $x^2+3x+4=0$ are complex roots , both roots are common.
For common roots,$\frac{a}{1}=\frac{b}{2}=\frac{c}{4}=\lambda$ (say)
$\Rightarrow a=\lambda ,b=3\lambda ,c=4\lambda$
$\therefore \frac{a+c}{b}=\frac{5\lambda }{3\lambda }=\frac{5}{3}\ne \frac{4}{3}$.Hence, assertion is not true.
If both roots of $a^{\prime }{x}^2+b^{\prime }x+c^{\prime }=0$ and $a^{\prime \prime }{x}^2+b^{\prime \prime }x+c^{\prime \prime }=0$ are identical,
then $\frac{a^{\prime }}{a^{\prime \prime }}=\frac{b^{\prime }}{b^{\prime \prime }}=\frac{c^{\prime }}{c^{\prime \prime }}$ where $a{,}^{\prime }b{,}^{\prime }c{,}^{\prime \prime }b{,}^{\prime \prime }c,\in R$ which is the condition for both roots common.
Hence, reason is true.