AP EAMCET Engineering 21 Sep 2020 Shift 1 Paper

Given vectors $u = \hat{i} - 2\hat{j}$ and $v = -3\hat{i} + 5\hat{j}$. Now, equation of line segment passes through the points $u$ and $v$ is $\frac{x-1}{4} = \frac{y+2}{-7}$....(i) Now, the point vectors having position vectors $O P = -\frac{1}{7}\hat{i}, O Q = -\frac{1}{4}\hat{j}$ and $O R = -2\hat{i} + 3\hat{j}$ are given. The point $P\left(-\frac{1}{7}, 0\right)$ by putting in line (i) $\because$ Point $P\left(-\frac{1}{7}, 0\right)$ satisfied the line, so it is on the line segment (i) The point $Q\left(0, -\frac{1}{4}\right)$ by putting in line (i) $\frac{0-1}{4} = \frac{-\frac{1}{4}+2}{7} \Rightarrow -\frac{1}{4} = -\frac{1}{4}$ $\because$ Point $Q\left(0, -\frac{1}{4}\right)$ also satisfied the line (i), so it lies on the line segment. Now, the point $R(-2,3)$, on putting in the equation of line (i) $\frac{-2-1}{4} \neq \frac{3+2}{-7}$ $\because$ Point $Q(-2,3)$ not satisfied the line (i), so it doesn't line on the line segment. Hence, option (a) is correct.