Question

Is the equation `13x² + 13y² - 26x + 52y = -78` a line, parabola, ellipse, hyperbola, or circle?

Answer 1

No - it contains no real points.

Explanation:

Given:

`13x^2+13y^2-26x+52y=-78`

First note that all of the coefficients are divisible by `13`, so let's divide the whole equation by `13` to get:

`x^2+y^2-2x+4y=-6`

Adding `1+4=5` to both sides of the equation and rearranging slightly, this becomes:

`x^2-2x+1+y^2+4y+4 = -1`

That is:

`(x-1)^2+(y+2)^2 = -1`

Note that this is always false for real values of `x` and `y`, so this equation describes an empty set of values in the `x` `y` plane.

Answer 2

Imaginary circle

Explanation:

The given equation:

`13x^2+13y^2-26x+52y=-78`

`13(x^2+y^2-2x+4y)=-78`

`x^2+y^2-2x+4y=-6`

`(x^2-2x+1)+(y^2+4y+4)-5=-6`

`(x-1)^2+(y+2)^2=-1`

`(x-1)^2+(y+2)^2=i^2`

Above equation shows a circle with a radius `r=i` i.e. an imaginary circle with unit radius.