What is the radius of the circle 4x² + 4y² - 20x + 12y - 15 = 0?

Concept:

The general form of the equation of a circle is:

x2 + y2 + 2gx + 2fy + c = 0

The centre of the circle is (-g, -f).

The radius of the circle is \(√{g^{2}+f^{2}-c}\).

Calculation:

Given equation of a circle is, 

4x2 + 4y2 - 20x + 12y - 15 = 0

x2 + y2 - 5x + 3y - 15/4 = 0

On comparing from the general equation of circle

2g = 5, or g = 5/2

2f = 3 or f = 3/2 & c = -15/4

By using the above formula

Radius = \(√{\frac{25}{4}+\frac{9}{4}-(-\frac{15}{4})}\)

Radius = \(√{\frac{25+9+15}{4}} =√ \frac{49}{4} \) = √12.25

Radius = 3.5

Alternate Method

Given equation of a circle is, 

4x2 + 4y2 - 20x + 12y - 15 = 0

⇒ [4x2 - 20x] + [4y2 + 12y] - 15 = 0

⇒ [(2x)² - 2 . 2x . 5 + 5² - 5²] + [(2y)2 + 2 . 2y . 3 + 32 - 32] - 15 = 0

⇒ [(2x - 5)2 - 52] + [(2y + 3)2 - 32] - 15 = 0

⇒ (2x - 5)2  + (2y + 3)2 - 15 - 25 - 9 = 0

⇒ (2x - 5)² + (2y +3)² = 49

⇒ 4(x - \(5\over2\))² + 4(y + \(3\over2\))² = 49

⇒ (x - \(5\over2\))2 + (y + \(3\over2\))² = (\(7\over2\))²

∴ The radius of the circle = \(7\over2\)units = 3.5 units