# Point and Circle

Let S be a circle with center C and radius r (> 0) and P be any point in the plane of the circle S, then

- P is called
**exterior**to S iff | C P | > r, - P is called
**interior**to S iff |CP| < r and - P is said to
**lie**on S iff |CP| = r.

If P is **exterior** to S then we say that P lies **outside** S, and
if P is **interior** to S then we say that P lies **inside** S

**Notation.** Let S = x² + y² +2 g x +2 f y + c = 0, g² + f² - c > 0, be
a circle and P (x_{1}, y_{1}) be a point in the plane of S,
then S_{1} = x_{1}² +y_{1}² +2 g x_{1} +2 f y_{1} + c.

Let S be a circle and P (x_{1}, y_{1}) be a point in the plane of S, then

- P is exterior to S iff S
_{1}> 0 - P is interior to S iff S
_{1}< 0 - P lies on S iff S
_{1}= 0

**Corollary.**

Let S be a circle and P (x_{1}, y_{1}), Q (x_{2}, y_{2}) be two
points in the plane of S then they lie

- on the same side of S iff S
_{1}and S_{2}have same sign - on the opposite sides of S iff S
_{1}and S_{2}have opposite signs.

## Exercise

- Among the points given below, point out which of these are interior,
exterior or lie on the circle S with center (1, 2) and radius 3?

(i) (2, 3) (ii) (-1, -3) (iii) (4, 2) - Among the points given below, point out which of these are exterior,
interior or lie on the circle 3 x 2 +3 y 2 -2 x -11 = 0?

(i) (-1, 0) (ii) (1, -2) (iii) - Do the following pairs of points lie on the same side or on opposite
sides of the circle with center (- 1, 2) and radius?

(i) (2, -3) and (1, 2) (ii) (0, 4) and (1, 3)

## Answers

**1.**(i) Interior (ii) exterior (iii) lies on

**2.**(i) Interior (ii) exterior (iii) lies on

**3.**(i) Opposite sides (ii) both lie on the circle