Question

Which equation is equivalent to the following?

Answer 1

Answer:

4)  2(x - 3)^2 + 2(y + 5)^2 = 96.

Step-by-step explanation:

2x^2 – 12x + 2y^2 + 20y – 28 = 0

2(x^2 – 6) + 2(y^2 + 10y) = 28

2 ( (x - 3)^2 - 9)) + 2((y + 5)^2 - 25) = 28

2(x - 3)^2 - 18 + 2(y + 5)^2  - 50 = 28

2(x - 3)^2 + 2(y + 5)^2 = 28 + 18 + 50

2(x - 3)^2 + 2(y + 5)^2 = 96

Answer 2

Answer:

$4) 2(x-3)^2+2(y+5)^2=96$

Step-by-step explanation:

$2x^2 -12x +2y^2 + 20y - 28 = 0$

It basically represents a circle.

Equation of circle

$\left(x-a\right)^2+\left(y-b\right)^2=r^2\:$

$2x^2-12x+2y^2+20y=28$

$\frac{1}{2} (2x^2-12x+2y^2+20y)=28 \cdot \frac{1}{2}$

$x^2-6x+y^2+10y=14$

Organize the variables

$\left(x^2-6x\right)+\left(y^2+10y\right)=14$

Now we have to complete the square for both groups:

$\left(x^2-6x+9\right)+\left(y^2+10y+25\right)=14+9+25$

$\left(x-3\right)^2+\left(y+5\right)^2=14+9+25$

$\left(x-3\right)^2+\left(y+5\right)^2=48$

Now, I would proceed with:

$\sqrt{48} =\sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} =4\sqrt{3}$

$\left(x-3\right)^2+\left(y+5\right)^2=(4\sqrt{3} )^2$

We got the center and the circle radius

$\left(3,\:-5\right)\\r=4\sqrt{3}$

But this is not what you want:

So, just multiply both sides by 2 and you'll get:

$2\left(x-3\right)^2+2\left(y+5\right)^2=96$