MATHEMATICS

The circle
The equation of the tangent at P(4, 3) is
y − 3 y 3
4
y= −
4
= − x 4
3 ( − = x− x
3 ( −
3 4) ( − 4) 4)
3y
➝
− 93y= − 16 39y−
= − 416 9x
= −16 4x−
4x
➝ 4x
+ 34yx− + 25 43xy=
+ − 0325 y − = 25 0 = 0
(ii) OQR forms a right-angled triangle.
Find Q:
3 y
− 25 =
0
➝ y =
25
3
Find R:
4 x
− 25 =
0
➝
x
=
25
4
Area of triangle OQR is
1 25
2
× 25 625
4
× 3
= 24
square units.
Area is 1 × base × height.
2
The base is the x coordinate
of R and the height is the
y coordinate of Q.
Substitute x = 0 into the
tangent equation to find Q.
Substitute y = 0 into the
tangent equation to find R.
Exact means leave your answer
as a fraction (or a surd).
Exercise 5.4
1 Find the equations of the following circles.
(i) centre (2, 3), radius 1
(ii) centre (2, −3), radius 2
(iii) centre (−2, 3), radius 3
(iv) centre (−2, −3), radius 4
2 For each of the following circles state
(a) the coordinates of the centre
(b) the radius.
(i) x 2 + y 2 = 1
(ii) x 2 + (y − 2) 2 = 2
(iii) (x – 2) 2 + y 2 = 3
(iv) (x + 2) 2 + (y + 2) 2 = 4
(v) (x − 2 ) 2 + (y +2) 2 = 5
3 The equation of a circle is (x − 3) 2 + (y + 2) 2 = 26.
Complete the table to show whether each point lies inside the circle,
outside the circle or on the circle.
Point Inside Outside On
(3, −2) ¸
(−2, −5)
(6, −6)
(4, 3)
(0, 2)
(−2, −3)
22