The equation of the line passing through the points $(1, 2)$ and $(2, 5)$ is:

The points $A (1, 3)$ and $C (5, 1)$ are the opposite vertices of a rectangle. The equation of the line passing through the other two vertices and having a gradient of $2$ is:

Suppose a line makes an angle of $120^{\circ}$ with the positive direction of $X$-axis. If the length of the perpendicular from the origin to that line is $4$,then the equation of the line is

If the line passing through the point $(4, -3)$ and having a negative slope makes an angle of $45^{\circ}$ with the line joining the points $(1, 1)$ and $(2, 3)$,then the sum of the intercepts of that line is:

The polar equation $\cos \theta + 7 \sin \theta = \frac{1}{r}$ represents a

Let $u = \hat{i} - 2\hat{j}$ and $v = -3\hat{i} + 5\hat{j}$. Consider three points $P, Q,$ and $R$ having position vectors $-\frac{1}{7}\hat{i}$,$-\frac{1}{4}\hat{j}$,and $-2\hat{i} + 3\hat{j}$ respectively. Among these,the points on the line segment passing through $u$ and $v$ are