Equation of a circle questions. please help?

Question

hi, please help me with these 2 problems. thanks.

Find the radius and the coordinates of the center of the circle for each question.
4(x-3)^2 + 4(y-2)^2=10

Write the standard form of the equation.
x^2 + y^2 + x = 3/4

Answer 1

1) The circle equation is 4(x - 3)² + 4(y - 2)²  = 10.

The standard form of the circle equation is ( x - h )² + ( y - k )² = r², where, (h, k) is the center of the circle, and r is the radius.

Write the equation in standard form of a circle.

Here x ² and y ²coefficient is 4 , for perfect square make x ²and y ² coefficient 1 by dividing each side by 4.

[4(x - 3)²]/4 + [4(y - 2)²]/4  = 10/4

(x - 3)²+ (y - 2)²  = 5/2

(x - 3)²+ (y - 2)²  = [√(5/2)]²

Compare the equation with standard form of a circle equation.

The center (h, k) is (3, 2), and

The radius (r) is √(5/2) units.

Answer 2

2) The circle equation is x² +  y² + x = 3/4.

x² + x + y²  = 3/4.

The standard form of the circle equation is ( x - h )² + ( y - k )² = r², where, (h, k) is the center of the circle, and r is the radius.

Write the equation in standard form of a circle.

To change the expression into a perfect square  add (half the x coefficient)² and add (half the y coefficient)²to each side of the expression.

Here, x coefficient = 1, so, (half the x coefficient)² = (1/2)²= 1/4.

Here, y coefficient = 0, so, (half the y coefficient)² = (0)²= 0.

Add 1/4 and 0 to each side.

x² +  x + 1/4 + y² + 0 = 3/4 + 1/4 + 0

(x + 1/2)² + (y - 0)²  = 1

[x - (- 1/2)]² + (y - 0)²  = 1²

Compare the equation with standard form of a circle equation.

The center (h, k) is (-1/2, 0), and

The radius (r) is 1 unit.