This conic appears at first to be an ellipse since there is
no xy term, an x² and a y² term with different coefficients
with the same sign when on the same side of the equation.
So we try to get it in standard form. Either
in the form 
or 
4x² + 5y² - 8x + 20y = -24
4x² - 8x + 5y² + 20y = -24
4(x² - 2x) + 5(y² + 4y) = -24
To complete the square in the first parentheses,
Multiply the coefficient of x, which is -2, by 1/2
getting -1. Then square -1, which gives +1. So we
add +1 to the right of the first parentheses, and
since there is a 4 multiplied by the first parentheses
we add 4·1 or 4 to the right side of the equation:
4(x² - 2x + 1) + 5(y² + 4y) = -24 + 4
To complete the square in the second parentheses,
Multiply the coefficient of y, which is 4, by 1/2
getting 2. Then square 2, which gives 4. So we
add 4 to the right of the second parentheses, and
since there is a 5 multiplied by the second parentheses
we add 5·4 or 20 to the right side of the equation:
4(x² - 2x + 1) + 5(y² + 4y + 4) = -24 + 4 + 20
Factor the first parentheses as (x - 1)(x - 1) or (x - 1)²
Factor the second parentheses as (x - 2)(x - 2) or (x - 2)²
Combine the numbers on the right, getting 0.
4(x - 1)² + 5(y + 2) = 0
Oh oh. This is an unusual situation because the right
side came out to be 0. So we cannot get 1 on the right
side as we could if it were some other number.
This means that the graph is of an ellipse that is
degenerated into a single point. The graph of the
equation is simply a single point (1,-2)
The graph is simply this single point (1,-2):
This equation is the equation of a single point. A single point
IS a conic section, because the vertex of a cone is a single point.
A plane intersecting a cone at its vertex is a single point.
Edwin
