2. Convert the given equation from standard to general form $$ (x-1)^{2}+(y+2)^{2}=81 $$

Question

UpStudy AI · Accepted Answer

To convert the given equation from standard form to general form, we start with the standard form of a circle:

$$
(x - 1)^2 + (y + 2)^2 = 81
$$

First, we will expand both squared terms.

1. Expand $$(x - 1)^2$$:
   $$
   (x - 1)^2 = x^2 - 2x + 1
   $$

2. Expand $$(y + 2)^2$$:
   $$
   (y + 2)^2 = y^2 + 4y + 4
   $$

Now, substitute these expansions back into the equation:

$$
x^2 - 2x + 1 + y^2 + 4y + 4 = 81
$$

Next, combine the constant terms on the left side:

$$
x^2 - 2x + y^2 + 4y + 5 = 81
$$

Now, we will move 81 to the left side of the equation:

$$
x^2 - 2x + y^2 + 4y + 5 - 81 = 0
$$

This simplifies to:

$$
x^2 - 2x + y^2 + 4y - 76 = 0
$$

Thus, the general form of the equation is:

$$
x^2 + y^2 - 2x + 4y - 76 = 0
$$
The general form of the equation is $$ x^2 + y^2 - 2x + 4y - 76 = 0 $$.

2. Convert the given equation from standard to general form \[ (x-1)^{2}+(y+2)^{2}=81 \]