Equation of a Circle

Edexcel IAL WMA12/01/P2 /Jan 2020/Q6 (Equation of Circles, Circles Geometry)

The circle C has equation​​ 

x2 + y2 + 6x – 4y – 14 = 0

• Find​​ 

• the coordinates of the centre of C,​​ 

• the exact radius of C.​​ 

(3)​​ 

The line with equation y = k, where k is a constant, is a tangent to C.​​ 

• Find the possible values of k.​​ 

(2)​​ 

The line with equation​​ y = p, where p is a negative constant, is a chord of C.​​ 

Given that the length of this chord is 4 units,​​ 

• find the value of p.​​ 

(3)

SOLUTION

a- ​​ Changing the equation to the standard circle equation.​​ 

x2+6x+y2-4y=14

Using the completing square method,​​ 

x2+6x+y2-4y=14

x+32-9+y-22-4=14

x+32+y-22=27

a-i-​​ 

Center-3, 2

a- ii- ​​ 

radius=27

r=33 

b- ​​ There are two possible ways for the tangent to exist. One is the tangent cuts at the top, and the other cuts at the bottom. ​​ 

Hence, the range of k is given as​​ 

k=2±3

c- Since​​ y=p​​ forms the chord of the circle and it is given that p is negative.​​ 

Using the pythagoras theorem to find the height of the traingle or x.

H2=P2+B2

332=x2+22

x=332-22

x=27-4

x=23  

Now, subtracting ​​ x-value form y-cordinate of ​​ centre which is 2, so that we may have the equation of chord as​​ y=2-23

Thus,​​ 

p=2-23