Exercises: Coordinate geometry of the circle and radians

Write down the equation of the circle with centre $(2,-3)$ and radius $5$.

Find the centre and radius of the circle $x^2 + y^2 - 6x + 4y - 12 = 0$.

1. a) Convert $\frac{5\pi}{6}$ radians to degrees.

1. b) A sector of a circle of radius $8$ cm has angle $\frac{\pi}{3}$. Find its arc length and area, in terms of $\pi$.

The points $A(1,2)$ and $B(7,10)$ are the endpoints of a diameter of a circle $c$.

1. a) Find the centre and radius of $c$.

1. b) Find the equation of $c$.

1. c) Determine whether the point $(8,4)$ lies inside, on or outside $c$.

The circle $x^2 + y^2 = 25$ and the line $x - 2y + 5 = 0$ are given.

1. a) Find the coordinates of the points where the line meets the circle.

1. b) Find the length of the chord they determine.

A circular sports pitch in Athlone has radius $40$ m. A sector of the pitch is reserved for training, with central angle $2.1$ radians.

1. a) Find the length of the arc bounding the sector.

1. b) Find the area of the sector.

1. c) Find the area of the triangle formed by the two radii and the chord, correct to the nearest square metre, and hence the area of the segment.

The circle $c$ has equation $x^2 + y^2 - 4x - 6y - 12 = 0$.

1. a) Find the centre and radius of $c$.

1. b) Show that the point $P(5,7)$ lies on $c$ and find the equation of the tangent to $c$ at $P$.

1. c) The tangent at $P$ meets the $x$-axis at $Q$. Find the length $|PQ|$ and verify that it equals the length of the tangent from $Q$ to $c$ computed using the centre and radius.

A circle $c$ passes through $A(1,5)$, $B(7,5)$ and $C(8,2)$.

1. a) Find the equation of $c$.

1. b) Find the equations of the tangents to $c$ that are parallel to the line $3x + 4y = 0$.

Two circles, $c_1: x^2 + y^2 = 20$ and $c_2: (x-6)^2 + y^2 = 20$, are given.

1. a) Find the points of intersection $A$ and $B$.

1. b) Find the angle $\angle AO_1B$, in radians correct to three decimal places, where $O_1$ is the centre of $c_1$.

1. c) Hence find the area of the region common to both circles, correct to one decimal place.