Question

Assertion :If the equation $x^2+bx+ca=0$ and $x^2+cx+ab=0$ have a common root, then their other root will satisfy the equation $x^2+ax+bc=0$ Reason: If the equation $x^2=bx+ca=0$ and $x^2+cx+ab=0$ have a common root, then $a+b+c=0$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $\alpha ,\beta$ be the roots of $x^2+bx+ca=0$ and

$\alpha ,\gamma$ be the roots of $x^2+cx+ab=0,$ then we have,

${\alpha }^2+b\alpha +c\alpha =0$ and ${\alpha }^2+c\alpha +ab=0$

Substrating, we have,

$(b-c)\alpha +a(c-b)=0\Rightarrow \alpha =a$

Putting $\alpha =a$ in equation $x^2+bx+ca=0,$ we have $a^2+ab+ca=0$

i.e., $a+b+c=0$ ...(1)

Also, we have

$\alpha \beta =ca$ and $\alpha \gamma =ab$

$\Rightarrow \beta =c$ and $\gamma =b$

Now, $\beta +\gamma =b+c$ and $\beta \gamma =bc.$

Hence, $\beta ,\gamma$ will be the roots of the equation

$x^2-(b+c)x+bc=0$

i.e., $x^2+ax+bc=0.$ ....$[$Using (1)$]$