Question

The distance between the centres of two circles.
x²+ y² = 4 and x²+ y² - 6y-8y=0 is​

Answer 1

Answer :

5 units

Note :

• The standard form of a circle is given by : (x - h)² + (y - k)² = r² , where (h , k) is the centre and r is the radius of the circle .
• The distance between the points A(x1 , y1) and B(x2 , y2) is given by : d = √[ (x2 - x1)² + (y2 - y1)² ]

Solution :

Here ,

The given equations of circles are :

x² + y² = 4 ------C

x² + y² - 6x - 8y = 0 -------C'

Now ,

The equation of circle C can be rewritten as ;

→ x² + y² = 4

→ (x - 0)² + (y - 0)² = 2²

Clearly ,

The centre of the circle C is O(0,0) and it's radius r = 2 .

Also ,

The equation of circle C' can be rewritten as ;

→ x² + y² - 6x - 8y = 0

→ x² - 6x + y² - 8y = 0

→ (x² - 6x + 3²) + (y² - 8y + 4²) = 3² + 4²

→ (x - 3)² + (y - 4)² = 9 + 16

→ (x - 3)² + (y - 4)² = 25

→ (x - 3)² + (y - 4)² = 5²

Clearly ,

The centre of the circle C' is O'(3,4) and it's radius r' = 5 .

Now ,

The distance between the centres of the two circles will be given as ;

→ d = OO'

→ d = √[ (3 - 0)² + (4 - 0)² ]

→ d = √[ 3² + 4² ]

→ d = √(9 + 16)

→ d = √25

→ d = 5

Hence ,

The required distance is 5 units .