Standard form of two equations. Hyperbolas?

Question

#1: 25X^2+9Y^2-150X+36Y+36=0
whats the standard form. center, ect?
and how do you come up with it?
please and thank you

Answer 1

This is an ellipse, but the process for putting it in standard form is the same:

25X^2 + 9Y^2 - 150X + 36Y + 36 = 0

25x^2 - 150x + 9y^2 + 36y = -36

25(x^2 - 6x) + 9(y^2 + 4y) = -36

25(x^2 - 6x + 9) + 9(y^2 + 4y + 4) = -36 + 225 + 36

25(x - 3)^2 + 9(y + 2)^2 = 225

[25(x - 3)^2]/225 + [9(y + 2)^2]/225 = 225/225

[(x - 3)^2]/9 + [(y + 2)^2]/25 = 1

The center is (3, -2)

The minor axis is 6 units long and horizontal

The major axis is 10 units long and vertical

Answer 2

Put the equation of the conic in standard form.

25x² + 9y² - 150x + 36y + 36 = 0
25x² - 150x + 9y² + 36y = -36

Complete the squares of the x terms and of the y terms.

25(x² - 6x) + 9(y² + 4y) = -36
25(x² - 6x + 9) + 9(y² + 4y + 4) = -36 + 25*9 + 9*4
25(x - 3)² + 9(y + 2)² = 225

Divide by 225.

(x - 3)²/9 + (y + 2)²/25 = 1

This is the equation of an ellipse with center (3, -2).

The semi-major axis is vertical and is of length 5.
The semi-minor axis is horizontal and is of length 3.