Question

If $ {{x}^{2}}+ax+10=0 $ and $ {{x}^{2}}+bx-10=0 $ have a common root, then $ {{a}^{2}}-{{b}^{2}} $ is equal to:

• A. 10
• B. 20
• C. 30
• D. 40
• E. 50

Answer 1

Answer: (D)

Let $ \alpha $ be the common root for both the equations $ {{x}^{2}}+ax+10=0 $ and $ {{x}^{2}}+bx-10=0 $ $ \therefore $ $ {{\alpha }^{2}}+a\alpha +10=0 $ ...(i) and $ {{\alpha }^{2}}+b\alpha -10=0 $ ...(ii) From Eqs. (i) and (ii), we get $ (a-b)\alpha +10+10=0 $ $ \Rightarrow $ $ (a-b)\alpha =-20 $ $ \Rightarrow $ $ \alpha =\frac{-20}{a-b} $ $ \because $ $ \alpha $ is also the root of $ {{x}^{2}}+bx-10=0 $ $ \therefore $ $ \frac{400}{{{(a-b)}^{2}}}+b\left( \frac{-20}{a-b} \right)-10=0 $ $ 40-2b(a-b)={{(a-b)}^{2}} $ $ \Rightarrow $ $ {{a}^{2}}+{{b}^{2}}-2{{b}^{2}}=40 $ $ \Rightarrow $ $ {{a}^{2}}-{{b}^{2}}=40 $