Find the Standard Equation of x^2+2y^2+6x-20y+53=0 and State if its a Parabola,Ellipse, Or Hyperbola

Question

Find the standard Equation of the QUESTION

Answer 1

Given equation x^2+2y^2+6x-20y+53 = 0

To complete squre add 6 to each side.

x^2+2y^2+6x-20y+53+6 = 0+6

x^2+2y^2+6x-20y+50+9 = 6

x^2+6x+9+2y^2-20y+50 = 6

x^2+6x+9+2(y^2-10y+25) = 6

(x+3)^2+2(y-5)^2 = 6

Divide to each side by 6.

((x+3)^2)/6+(2(y-5)^2)/6 = 6/6

((x+3)^2)/6+((y-5)^2)/3 = 1

(x-(-3))^2/(√6)^2+(y-5)^2/(√3)^2 = 1

Compare it to standard form of ellipse is (x-h)^2/a^2+(y-k)^2/k^2 = 1

Where a > b.

Given equation is ellipse.

Answer 2

The conic equation

To identify the conic section

General form of a conic equation in the form

Both variables are squared and have the same sign, but they aren't multiplied by the same number, so this is an ellipse.

is ellipse.

To find the ellipse in standard form.

To change the expressions (x ²+ 6x) and (y ² - 10y) into a perfect square trinomial,

add (half the x coefficient)² and add (half the y coefficient)² to each side of the equation

The standard form for an ellipse is in a form = 1, So divide both sides of equation by 6 to set it equal to 1.

Compare it to standard form of ellipse

a ² > b ²

If the larger denominator is under the "x " term, then the ellipse is horizontal.

center (h, k ) = (1, -2)

a  = length of semi-major axis .

b  = length of semi-minor axis .

Standard form of is .