Practicing Success
E and F are the interior points on the sides BC and CD of a parallelogram ABCD. Let $\vec{B E}= 4 \vec{EC}$ and $\vec{CF}=4 \vec{F D}$. If the line EF meets the diagonal AC in G then $\vec{A G}=\lambda \vec{A C}$ where $\lambda$ is equal to |
$\frac{21}{25}$ $\frac{1}{3}$ $\frac{7}{13}$ $\frac{21}{5}$ |
$\frac{21}{5}$ |
Let P.V. of A.B. and D be $\vec{0}, \vec{b}, \vec{d}$. Then $\vec{AC}=\vec{b}+\vec{d}$ $\Rightarrow \vec{E}=\frac{\vec{b}+4(\vec{b}+\vec{d})}{5}=\vec{b}+\frac{4}{5} \vec{d}$ and $\vec{F}=\frac{\vec{b}+\vec{d}+4 \vec{d}}{5}=\vec{d}+\frac{1}{5} \vec{b}$ Equation of EF : $\vec{r}=\vec{b}+\frac{4}{5} \vec{d}+\lambda\left(\frac{4}{5} \vec{b}-\frac{1}{5} \vec{d}\right)$ Equation of AC : $\vec{r}=\lambda_1(\vec{b}+\vec{d})$ For point G we must have, $\vec{b}+\frac{4}{5} \vec{d}+\frac{\lambda}{5}(4 \vec{b}-\vec{d})=\lambda_1(\vec{b}+\vec{d})$ $\Rightarrow \lambda_1=\frac{21}{5}, \lambda=-\frac{1}{5} \Rightarrow \vec{AG}=\frac{21}{5} \vec{AC}$ Hence (4) is correct answer. |