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

2 Answers
Jul 25, 2018

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.

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.