Circles & Parabolas — practice questions

Find the centre and radius of the circle $x^2 + y^2 + 8x - 6y - 11 = 0$, and state whether the point $(1, 2)$ lies inside, on, or outside it.

A circular irrigation pivot waters a field modelled by $(x - 6)^2 + (y - 4)^2 = 100$, in metres. A straight drainage channel runs along $3x + 4y = 20$. Determine whether the channel passes through the watered area, and find the length of channel inside it.

Find the equation of the circle passing through the three points $A(1, 1)$, $B(7, 3)$ and $C(4, -4)$, and prove that no circle passes through three collinear points.

For the parabola $(y - 2)^2 = -8(x + 1)$, state the vertex, the focus, the directrix, and the direction in which it opens.

A suspension footbridge has a parabolic cable. The two towers are 60 m apart and the cable is attached 12 m above the deck on each tower, sagging to 2 m above the deck at its lowest point. Find the equation of the cable, and the height of the cable 10 m from a tower.

Prove, from the focus–directrix definition, that the set of points equidistant from the point $(0, p)$ and the line $y = -p$ has equation $x^2 = 4py$. Then use the result to show that the latus rectum has length $|4p|$.