I do not understand how to do this circle question?

Question

x^2 + y^2 + 6x - 4y +3 =0

it asks to find the center and radius and the tangent line at point (-2,5)

im not used to the formula of a circle being in that format, please help

Answer 1

x² + y² + 6x - 4y + 3 = 0

rearrange the terms

x² + 6x + y² - 4y + 3 = 0

(x² + 6x) + (y² - 4y) + 3 = 0

We need to complete the square for each bracketted group. We will have to add some unknown number A to the first group to make it a perfect square that can be factored. And will will have to add some unknown number B to the second group to make it a perfect square that can be factored. To keep everything in balance, we have to add A and B to both sides.

(x² + 6x + A) + (y² - 4y + B) + 3 = 0 + A + B

For the first bracketted group, take 1/2 of 6 (=3), then square it (=9). So A = 9.

For the second bracketted group, take 1/2 of -4 (=-2), then square it (=4). So B = 4

(x² + 6x + 9) + (y² - 4y + 4) + 3 = 0 + 9 + 4

(x² + 6x + 9) + (y² - 4y + 4) + 3 = 13

Now that both bracketted groups are perfect square, we can factor each one.

(x + 3)² + (y - 2)² = 13 - 3

(x + 3)² + (y - 2)²  = (√10)²

The equation above is now in standard form. Now do you recognize it?

It tells us that the center is (-3,2) and the radius is √10.

See if you can figure out the tangent line at point (-2,5).