The equations x²-a x+b=0 and x²+b x-a=0 have a common root, then

Correct Answer is : a+b=0a+b=0a+b=0 or a−b=1a-b=1a−b=1

Correct Answer is : a+b=0a+b=0a+b=0 or a−b=1a-b=1a−b=1

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AP EAPCET 24-Aug-2021 Shift 1 Paper

Section: Mathematics

Question : 12 of 160

Marks: +1, -0

The equations

x^{2}-a x+b=0

x^{2}+b x-a=0

have a common root, then

Solution:

Equation

x^{2}-a x+b

and

x^{2}+b x-a

have

a

common root. Let

\alpha

be the common root of both quadratic equations

\Rightarrow \alpha^{2}-a \alpha+b=0

\alpha^{2}+b \alpha-a=0

Using cross-multiplication

\frac{\alpha^{2}}{\begin{array}{cc} -a & b \\ b & -a \end{array}} = \frac{\alpha}{\begin{array}{cc} b & 1 \\ -a & 1 \end{array}} = \frac{1}{\begin{array}{cc} 1 & -a \\ 1 & b \end{array}}

\frac{\alpha^{2}}{a^{2}-b^{2}} = \frac{\alpha}{b+a} = \frac{1}{b+a}

We get

\frac{\alpha}{b+a} = \frac{1}{b+a}

\Rightarrow \alpha=1

Now, using first two fraction

\frac{1}{a^{2}-b^{2}} = \frac{1}{a+b}

\Rightarrow a+b=a^{2}-b^{2}

\Rightarrow (a+b)=(a+b)(a-b)

\Rightarrow (a+b)(a-b-1)=0

a+b=0

a-b-1=0

\Rightarrow a-b=1

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