question_answer

Let \[P({{x}_{1}},{{y}_{1}})\] and \[Q({{x}_{2}},{{y}_{2}})\]are two points such that their abscissa \[{{x}_{1}}\] and \[{{x}_{2}}\] are the roots of the equation \[{{x}^{2}}+2x-3=0\] while the ordinates \[{{y}_{1}}\] and \[{{y}_{2}}\] are the roots of the equation\[{{y}^{2}}+4y-12=0\]. The centre of the circle with PQ as diameter is [Orissa JEE 2005]

A)            \[(-1,-2)\]                                  

B)            \[(1,\,\,2)\]

C)            \[(1,-2)\]                                    

D)            \[(-1,2)\]

Correct Answer: A

Solution :

           \[{{x}_{1}},{{x}_{2}}\]are roots of \[{{x}^{2}}+2x+3=0\]                    Þ \[{{x}_{1}}+{{x}_{2}}=-2\]                    \ \[\frac{{{x}_{1}}+{{x}_{2}}}{2}=-1\]                    \[{{y}_{1}},{{y}_{2}}\] are roots of \[{{y}^{2}}+4y-12=0\]                    Þ \[{{y}_{1}}+{{y}_{2}}=-4\Rightarrow \frac{{{y}_{1}}+{{y}_{2}}}{2}=-2\]                    Centre of circle\[\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right)=(-1,-2)\].