Show that the equation represents a circle by rewriting it in standard form. Then find the center and radius of the circle.

Question

Show that the equation represents a circle by rewriting it in standard form.

4x² + 4y² − 5x = 0

= ???

Find the center and radius of the circle.

(x,y)= (?,?)

r= ?

Answer 1

Put the x terms together. Divide by 4 to get x ^2-(5/4)x + y^2 =0

Take half the (5/4) = 5/8 and square it, to get 25/64

Add 25/65 to both sides to get

x^2 - (5/4)x + 25/64 + y^2 = 25/64

factor the 1st three terms, and rewrite the right side as a squared term

(x-5/8)^2 + y^2 = (5/8)^2

center is (5/8,0) radius is 5/8

standard form is (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r=radius

in this problem h=5/8, k=0 and r=5/8

Answer 2

4x² + 4y² - 5x = 0 (1)

The equation of the circle is in the general form:

x² + y² + 2gx + 2fy + c = 0

We want to write equation in terms of its center and radius:

(x - h)² + (y - k)² = r² (2)

4x² + 4y² - 5x = 0

Collect like terms:

(4x² - 5x) + 4y² = 0

Divide both sides by 4, to make the coefficients of x² and y² to be unity (i.e. 1):

x² - (5/4)x + y² = 0

Complete the square on both sides (i.e. add half the square of the coefficient of x on both sides of the equation):

x² - (5/4)x + (-5/4*1/2)² + y² = (-5/4*1/2)²

x² - (5/4)x + (-5/8)² + y² = (-5/8)²

Rewrite the coefficient of x as twice the square constant:

x² - 2(5/8)x + (-5/8)² + y² = (-5/8)²

Factoring gives:

(x - 5/8)(x - 5/8) + y² = (-5/8)²

(x - 5/8)² + y² = 25/64

Comparing with equation (2): (x - h)² + (y - k)² = r²

h = 5/8, k = 0, r = √(25/64) = 5/8

Center: (h, k) = (5/8, 0)

Radius: r = 5/8

Check:

(x - 5/8)² + y² = 25/64

Expand:

x² - 2(5/8)x + (-5/8)² + y² = 25/64

x² - 2(5/8)x + 25/64 + y² = 25/64

x² - (5/4)x + y² = 0

Multiply by 4 throughout:

4x² + 4y² - 5x = 0