Everything Maths Grade 12

9.3 CHAPTER 9. GEOMETRY

The distance between the two points is:

r =

1

2

PQ =

1

2

�

(x 2 −x 1 )^2 + (y 2 −y 1 )^2

=

1

2

�

(5− (−5))^2 + (− 5 − 5)^2

=

1

2

�

(10)^2 + (−10)^2

=

1

2

√

100 + 100

=

�

200

4

=

√

50

Step 4 : Write the equation of the circle

x^2 +y^2 = 50

Example 4: Equation of a Circle II

QUESTION

Find the centre and radius of the circle

x^2 − 14 x +y^2 + 4y =− 28.

SOLUTION

Step 1 : Change to standard form

We need to rewrite theequation in the form (x−x 0 ) + (y−y 0 ) = r^2

To do this we need to complete the square

i.e. add and subtract (^12 cooefficient of x)^2 and (^12 cooefficient of y)^2

Step 2 : Adding cooefficients

x^2 − 14 x +y^2 + 4y =− 28

∴ x^2 − 14 x + (7)^2 − (7)^2 +y^2 + 4y + (2)^2 − (2)^2 =− 28

Step 3 : Complete the squares

∴ (x− 7)^2 − (7)^2 + (y + 2)^2 − (2)^2 =− 28

Step 4 : Take the constants to the other side

∴ (x− 7)^2 − 49 + (y + 2)^2 − 4 =− 28

∴ (x− 7)^2 + (y + 2)^2 =−28 + 49 + 4

∴ (x− 7)^2 + (y + 2)^2 = 25

Step 5 : Read the values from the equation