9.3 CHAPTER 9. GEOMETRY
Now, we know that thetangent passes through (x 1 ;y 1 ) so the equation is givenby:
y−y 1 = mg(x−x 1 )y−y 1 =−1
mf
(x−x 1 )y−y 1 =−1
y 1 −y 0
x 1 −x 0(x−x 1 )y−y 1 =−x 1 −x 0
y 1 −y 0
(x−x 1 )For example, find the equation of the tangent tothe circle at point (1; 1). The centre of the circle is at
(0; 0). The equation of the circle is x^2 +y^2 = 2.
Use
y−y 1 =−
x 1 −x 0
y 1 −y 0
(x−x 1 )with (x 0 ;y 0 ) = (0; 0) and (x 1 ;y 1 ) = (1; 1).
y−y 1 =−
x 1 −x 0
y 1 −y 0(x−x 1 )y− 1 =−1 − 0
1 − 0
(x− 1)y− 1 =−1
1
(x− 1)
y =−(x− 1) + 1
y =−x + 1 + 1
y =−x + 2Exercise 9 - 8
- Find the equation ofthe circle:
(a) with centre (0; 5) and radius 5
(b) with centre (2; 0) and radius 4
(c) with centre (5; 7) and radius 18
(d) with centre (−2; 0) and radius 6
(e) with centre (−5;−3) and radius√
3
- (a) Find the equation of the circle with centre (2; 1) which passes through (4; 1).
(b) Where does it cut the line y = x + 1?
(c) Draw a sketch to illustrate your answers. - (a) Find the equation of the circle with centre (−3;−2) which passes through (1;−4).
(b) Find the equation ofthe circle with centre (3; 1) which passes through (2; 5).
(c) Find the point wherethese two circles cut each other. - Find the centre and radius of the following circles:
(a) (x− 9)^2 + (y− 6)^2 = 36
(b) (x− 2)^2 + (y− 9)^2 = 1
(c) (x + 5)^2 + (y + 7)^2 = 12
(d) (x + 4)^2 + (y + 4)^2 = 23
(e) 3(x− 2)^2 + 3(y + 3)^2 = 12