Question

Assertion :If the roots of the equations $x^2-bx+c=0$ and $x^2-cx+b=0$ differ by the same quantity, then $b+c$ is equal to $-4$ Reason: If $\alpha ,\beta$ are the roots of the equation $A{x}^2+Bx+C=0$, then $\alpha -\beta =\frac{\sqrt{B^2-4AC}}{A}$

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Reason:

We know that if $\alpha ,\beta$ are roots of the equation $A{x}^2+Bx+C=0$,

then $\alpha +\beta =-\frac{B}{A}$ and $\alpha \beta =\frac{C}{A}$

Then from ${(x-y)}^2={(x+y)}^2-4xy$

we get $\alpha -\beta =\frac{\sqrt{B^2-4AC}}{A}$

Assertion: Using reason and equating the value of $\alpha -\beta$ from both the given equations, we get

$\sqrt{b^2-4c}=\sqrt{c^2-4b}\Rightarrow b^2-4c=c^2-4b$

$\Rightarrow b^2-c^2=-4(b-c)\Rightarrow (b-c)(b+c+4)=0$

$\Rightarrow b+c=-4(\because b\ne c)$