Question

Find the centre and radius of the circle.
x² + y² + 8x + 10y - 8 = 0

Answer 1

See below ~

Explanation:

Given

• x² + y² + 8x + 10y - 8 = 0

Completing the square method (for both x and y)

1. Add and subtract 16 and 25.

• x² + 8x + 16 - 16 + y² + 10y + 25 - 25 - 8 = 0
• (x + 4)² + (y + 5)² - 49 = 0

2. Bring the constant term to the other side by adding 49 on both sides.

• (x + 4)² + (y + 5)² - 49 + 49 = 0 + 49
• (x + 4)² + (y + 5)² = 49

3. Centre = (x, y)

• x + 4 = 0
• x = -4
• y + 5 = 0
• y = -5
• Centre = (-4, -5)

4. Radius

• r² = 49
• r = √49
• radius = 7

Answer 2

Centre = ( - 4, - 5 )

Radius = 7

Explanation:

x² + y² + 8x + 10y - 8 = 0

It is in the form of x² + y² + 2gx + 2fy + c = 0 ( General circle equation ).

where,

2g = 8

g = 4

2f = 10

f = 5

c = - 8

Formula : -

The centre of the circle = ( - g, - f )

So,

The centre of the circle is ( - 4, - 5 ).

Formula : -

( Radius )² = g² + f² - c

( Radius )² = ( - 4 )² + ( - 5 )² - ( - 8 )

= 16 + 25 + 8

( Radius )² = 49

Square root on both sides,

Radius = √49 = 7