Conic sections identification — how to tell parabola from ellipse from hyperbola

Question

Given the general second-degree equation $ax^2 + bxy + cy^2 + dx + ey + f = 0$ , how do we identify whether it represents a circle, parabola, ellipse, or hyperbola?

(CBSE 11 + JEE Main — frequently asked in first shift)

Solution — Step by Step

For the equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ , the discriminant $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$ determines if the equation represents a conic (when $\Delta \neq 0$ ).

But for JEE Main, most questions give equations without the $xy$ term ( $h = 0$ ). Then the identification is simpler.

Compare the coefficients of $x^2$ and $y^2$ :

Condition	Conic
Only $x^2$ or only $y^2$ (one is missing)	Parabola
Both present, coefficients are equal and same sign	Circle
Both present, coefficients are different but same sign	Ellipse
Both present, coefficients have opposite signs	Hyperbola

$x^2 + y^2 - 6x + 4y - 12 = 0$ — coefficients of $x^2$ and $y^2$ are both 1 (equal, same sign). Circle.
$y^2 = 8x$ — only $y^2$ present, no $x^2$ . Parabola.
$\frac{x^2}{25} + \frac{y^2}{9} = 1$ — both $x^2$ and $y^2$ with different positive coefficients ( $\frac{1}{25}$ and $\frac{1}{9}$ ). Ellipse.
$\frac{x^2}{16} - \frac{y^2}{9} = 1$ — $x^2$ positive, $y^2$ negative. Hyperbola.

Each conic has a characteristic eccentricity $e$ :

Conic	Eccentricity
Circle	$e = 0$
Ellipse	$0 < e < 1$
Parabola	$e = 1$
Hyperbola	$e > 1$

flowchart TD
    A["Given: ax² + by² + ... = 0"] --> B{"Is there an xy term?"}
    B -- Yes --> C["Use discriminant: b² - 4ac"]
    C --> D{"b² - 4ac value?"}
    D -- "= 0" --> E["Parabola"]
    D -- "Negative" --> F["Ellipse or Circle"]
    D -- "Positive" --> G["Hyperbola"]
    B -- No --> H{"Coefficient of x² vs y²"}
    H -- "One is missing" --> E
    H -- "Equal, same sign" --> I["Circle"]
    H -- "Different, same sign" --> J["Ellipse"]
    H -- "Opposite signs" --> G

Why This Works

Conic sections are literally cross-sections of a cone cut by a plane at different angles. A horizontal cut gives a circle. A tilted cut gives an ellipse. A cut parallel to the slant gives a parabola. A steep cut (through both halves of the double cone) gives a hyperbola.

Algebraically, the relationship between the $x^2$ and $y^2$ coefficients reflects this geometry. Equal coefficients mean the curve extends equally in both directions (circle). Missing one term means the curve opens infinitely in one direction (parabola). Opposite signs mean two separate branches (hyperbola).

Alternative Method

For standard form equations, match the pattern directly:

Circle: $(x - h)^2 + (y - k)^2 = r^2$
Parabola: $(y - k)^2 = 4a(x - h)$ or $(x - h)^2 = 4a(y - k)$
Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
Hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

In JEE Main, if the question asks “which conic does this equation represent?”, first check for the $xy$ term. If absent, compare $x^2$ and $y^2$ coefficients — the answer takes 10 seconds. If the $xy$ term is present, use $b^2 - 4ac$ where the equation is $ax^2 + bxy + cy^2 + \ldots$

Common Mistake

Students sometimes identify $3x^2 + 3y^2 + 6x - 12y + 5 = 0$ as an ellipse because the equation “looks complicated.” But dividing through by 3 gives $x^2 + y^2 + 2x - 4y + \frac{5}{3} = 0$ — equal coefficients for $x^2$ and $y^2$ , so it is a circle. Always simplify first by dividing by the common factor.