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

The equation of the line,passing through the point $(a \cos^3 \theta, a \sin^3 \theta)$ and perpendicular to the line $x \cos \theta - y \sin \theta = a$,is

The equation of the line passing through the point $(1, 2)$ and perpendicular to the line $x + y + 1 = 0$ is

If $(0, 0)$,$(-2, 1)$,and $(5, 2)$ are the vertices of a triangle,then the equation of the line passing through the centroid and parallel to the line $x - 2y = 6$ is:

Reduce the equation $\sqrt{3} x + y - 8 = 0$ into normal form. Find the values of $p$ and $\omega$.

Reduce the equation $y-2=0$ into the normal form $x \cos \omega + y \sin \omega = p$. Find the perpendicular distance from the origin $(p)$ and the angle between the perpendicular and the positive $x$-axis $(\omega)$.