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Question

Circle: 4x^2+24x+4y^2+32y=58

Center:(___,___) (exact)

Radius:____ (exact)

Answer 1

4x² +  24x + 4y² + 32y = 58.

Divide each side by 4.

x² +  6x + y² + 8y = 29/2

It can be written as x² + 2(x)(3) + 3² - 3² + y² + 2(y)(4) + 4²- 4²= 29/2

MATHEMATICAL FORMULAE: (a + b)² = a² + 2ab + b²

So, (x + 3)² + (y + 4)² - 9 - 16 = 29/2

(x + 3)² + (y + 4)² - 25 = 14.5.

Add 25 to each side.

(x + 3)² + (y + 4)² = 39.5

(x + 3)² + (y + 4)² = (√39.5)².

Compare equation with standard from(x - h)² + (y - k)² = r² and here center (h, k), radius is r.

So, center (-3, -4) and radius r = √(39.5).

Answer 2

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.

The circle equation is 4x² +24x + 4y² + 32y = 58.

Write the equation in standard form of a circle.

To change the expression into a perfect square, first divide the given equation by 4.

x² +6x + y² + 8y = 29/2

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 = 6, so, (half the x coefficient)² = (6/2)²= 9.

Here y coefficient = 8, so, (half the y coefficient)² = (8/2)²= 16.

Add 9 and 16 to each side.

x² + 6x + 9 + y² + 8y + 16 = 29/2 + 9 + 16

(x + 3)² + (y + 4)² = (29 + 18 + 32)/2

(x + 3)² + (y + 4)² = 79/2

(x + 3)² + (y + 4)² = 39.5

(x - (- 3))² + (y - (- 4))² = (√(39.5))²

Compare the equation with standard form of a circle equation.

The center of the circle : (h, k) = (- 3, - 4), and

The radius : r = √(39.5) units.