question_answer

A parabola \[y=a{{x}^{2}}+bx+c\] crosses the X-axis at \[\left( \alpha +0 \right)\], \[\left( \beta +0 \right)\] both to the right of the origin. A circle also passes through those two points. The length of a tangent from the origin to the circle is

A) \[\sqrt{\frac{bc}{a}}\]

B) \[a{{c}^{2}}\]

C) \[\frac{b}{a}\]

D) \[\sqrt{\frac{c}{a}}\]

Correct Answer: D

Solution :

\[\alpha ,\beta \] are the roots of \[a{{x}^{2}}+bx+c=0\]

\[\therefore \,\,\,\,\alpha +\beta =-b/a\] and  \[\alpha \beta =c/a\]

Circle passes through \[\left( \alpha ,0 \right)\] and \[\left( \beta ,0 \right)\], \[OA=\alpha ,OB=\beta \]

\[\because \]   OT is tangent of circle

\[\therefore \,\,O{{T}^{2}}=OA.OB\] \[\Rightarrow \,\,O{{T}^{2}}=\alpha \beta \] \[\Rightarrow \,\,OT=\sqrt{\alpha \beta }\] \[\Rightarrow \,\,OT=\sqrt{c/a}\]