SHOW THAT THE EQUATION DOES NOT REPRESENTS A CIRCLE

Problem 1 :

Explain why x² + y² + 6x - 4y + 17 = 0 looks like it represents a circle but, in fact does not.

Solution :

x² + y² + 6x - 4y + 17 = 0

Finding radius of circle.

Comparing with general form

2g = 6, g ==> 3

2f = -4, f ==> -2

c = 17

Radius = √g² + f² - c

Radius = √3² + (-2)² - 17

Radius = √(9 + 4 - 17)

Radius = √-4

Since the radius is unreal, the given equation will not represent the circle.

Problem 2 :

Explain why x² + y² + 2x + 3y + 5 = 0 looks like it represents a circle.

Solution :

x² + y² + 2x + 3y + 5 = 0

Finding radius of circle.

Comparing with general form

2g = 2, g ==> 1

2f = 3, f ==> 3/2

c = 5

Radius = √g² + f² - c

Radius = √1² + (3/2)² - 5

Radius = √(1 + (9/4) - 5)

Radius = √(13/4) - 5

Radius = √(-7/4)

Since the radius is unreal, the given equation will not represent the circle.

Problem 3 :

Show that the equation x² + y² - 3x + 3y + 10 = 0 does not represent the circle.

Solution :

x² + y² - 3x + 3y + 10 = 0

Finding radius of circle.

Comparing with general form

2g = -3, g ==> -3/2

2f = 3, f ==> 3/2

c = 10

Radius = √g² + f² - c

Radius = √(-3/2)² + (3/2)² - 10

Radius = √(9/4) + (9/4) - 10

Radius = √(18/4) - 10

Radius = √(-22/4)

= √(-11/2)

Since the radius is unreal, the given equation will not represent the circle.