4

The equation $x^2+24xy+68y^2=0$ represents a

  1. Ellips

  2. Parabola

  3. Hyperbola

  4. Can't be decided

I know the general equations of all of these geometric figures, but I can't rework the given equation to match any of them. Completing the square w.r.t. $x$, I get $$(x+12y)^2-76y^2 = 0,$$

but it's still not on any desired form.

SvanN
  • 2,307
Parseval
  • 6,413

1 Answers1

5

Completing the square w.r.t $x$ i get $$(x+12y)^2-76y^2 \color{red}{=0}$$

This is a good step; now rewrite using $a^2-b^2 = (a-b)(a+b)$ to get: $$\left(x+12y + \sqrt{76}y \right)\left(x+12y - \sqrt{76}y \right)=0$$ But this implies: $$x+12y + \sqrt{76}y = 0 \;\vee\; x+12y - \sqrt{76}y = 0$$ And these are equations of...?

StackTD
  • 27,903
  • 34
  • 63
  • Straight lines? $x+y(12+\sqrt{76})=0\Leftrightarrow y=-\frac{1}{12+\sqrt{76}}x.$ – Parseval Aug 21 '17 at 09:48
  • @Parseval Indeed: two straight lines, intersecting in the origin. It is a so called degenerate conic. – StackTD Aug 21 '17 at 09:48
  • But the correct answer is hyperbola. – Parseval Aug 21 '17 at 09:50
  • @Parseval Yes, two intersecting lines is the degenerate form of a hyperbola. – StackTD Aug 21 '17 at 09:52
  • What is meant by degenerate and non degenerate? What is the difference between the two? – Parseval Aug 21 '17 at 09:52
  • @Parseval You can look up the definitions (see link above). Without getting technical: non degenerate refers to "real conics", i.e. a proper ellipse, parabola or hyperbola; the degenerate ones refer to the special cases such as intersecting lines (e.g. $x^2-y^2=0$) and parallel lines (e.g. $x^2=1$). – StackTD Aug 21 '17 at 09:55