question_answer

The length of a line segment AB is 2 units. It is divided into two parts at the point C such that \[A{{C}^{2}}=AB\times CB.\]What is the length of CB?

A)  \[3+\sqrt{5}\]units

B)  \[3-\sqrt{5}\]units

C)  \[2-\sqrt{5}\]units          

D)  \[\sqrt{3}\]units

Correct Answer: B

Solution :

Given, \[A{{C}^{2}}=AB\times CB\]

\[\Rightarrow \]   \[{{x}^{2}}=2\times \,\,(2-x)\]

\[\Rightarrow \]   \[{{x}^{2}}=4-2x\]

\[\Rightarrow \]   \[{{x}^{2}}+2x-4=0\]

\[\Rightarrow \]   \[x=\frac{-\,\,2\pm \sqrt{4+16}}{2\times 1}\]

\[\Rightarrow \]   \[x=-1\pm \sqrt{5}\]

Now, \[BC=2-(-1\pm \sqrt{5})\]

\[=3-\sqrt{5}\]   (neglect\[3+\sqrt{5}\because 3+\sqrt{5}>2\])