question_answer

Let \[A\,({{x}_{1}},{{y}_{1}})\]and \[B\,({{x}_{2}},{{y}_{2}})\] be any two points on the parabola \[y=a\,{{x}^{2}}+bx+c\] and let \[C\,({{x}_{3}},{{y}_{3}})\] be the point on the arc AB where the tangent is parallel to the chord AB. What is the value of \[{{x}_{3}}\] in terms of \[{{x}_{1}}\] and\[{{x}_{2}}\]?

A)  \[\frac{{{x}_{1}}-{{x}_{2}}}{2}\]        

B)  \[\frac{{{x}_{1}}+{{x}_{2}}}{2}\]     

C)  \[\frac{2{{x}_{1}}+3{{x}_{2}}}{2}\]    

D)  \[\frac{{{x}_{1}}+2{{x}_{2}}}{3}\]