Equation of a Circle

Edexcel IAL WMA12/01/P2 /Oct 2021/Q6 (Coordinates Geometry Circles)

• The circle C1​​ has equation​​ 

x2 + y2 + 10x - 12y = k where k is a constant

• Find the coordinates of the centre of C1​​ 

(2)

• State the possible range in values for k.​​ 

(2)

• The point P( p, 0), the point Q(−2,  10) and the point R(8,  −14) lie on a different circle, C2​​ 

Given that​​ 

•  p is a positive constant​​ 

• QR is a diameter of C2​​ 

find the exact value of p.​​ 

(4)

SOLUTION​​ 

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

x2+10x+y2-12y=k

Using the completung square method,​​ 

x2+10x+y2-12y=k

x+52-25+y-62-36=k

x+52+y-62-61=k

x+52+y-62=k+61

i-a-​​ Center ( - 5, 6) 

i- b- ​​ For the possible range of k,​​ 

r2=k+61

r=k+61

Remember,​​ k+61>0​​ otherwise the term inside the quare root would be negative.​​ 

k+61>0

k>-61

ii- ​​ Since QR is a diameter of C2 centre would lie at the midpoint of QR

Center -2+82, 10-142

C3, -2

Whereas, QC would be radius​​ 

QC=r =-52+122 

r=169 

r=13

Now, the equation of the circle will be​​ 

x-32+y+22=169

Since point P (p,0) lies on the circle, it must satify the equation of​​ C2.

x-32+22=169

x-32=165

x-3= ±165

x=3±165

It is given that p is positive constant.​​ 

p=3+165