# conics!!!!!!!!!!!!!!!!!!!!!!!

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

Center:(___,___) (exact)

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.

(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).

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.