Conic Sections — Identification & Eccentricity

A conic is the locus of a point $P$ for which $\dfrac{PF}{PM} = e$, where $F$ is the focus, the line is the directrix, $PM$ is the perpendicular distance to it, and $e\ge 0$ is the eccentricity. The value of $e$ classifies the curve:

• $e = 0$: circle
• $0 < e < 1$: ellipse
• $e = 1$: parabola
• $e > 1$: hyperbola

The latus rectum is the focal chord perpendicular to the axis; it recurs in every conic's formulas. Coordinate-geometry tools every conic question borrows (they are not conic facts, but nothing here works without them): the distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ for a chord or focal length; the perpendicularity test — two directions are perpendicular iff their dot product is 0, i.e. slopes multiply to $-1$; the section and midpoint formulas; and solving a quadratic (a parameter like $b$ in $b^2+b-2=0$ is fixed by factorising, with the negative root rejected when it is a length).

From the standard forms:

• Ellipse $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$ ($a>b$): $c^2 = a^2 - b^2$, $e = \tfrac{c}{a} < 1$.
• Parabola: $e = 1$ always.
• Hyperbola $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$: $c^2 = a^2 + b^2$, $e = \tfrac{c}{a} > 1$.

A minus sign between the squared terms (after normalising to $=1$) signals a hyperbola; a plus sign with unequal denominators signals an ellipse. Shared-foci conditions equate the two $c$ values.

For an equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$ (no $xy$ term), complete the square in $x$ and $y$:

• Equal positive coefficients on the squares → circle; unequal same-sign → ellipse; opposite signs → hyperbola; one square missing → parabola.
• Degenerate cases: if completing the square gives $(x-h)^2 + k(y-j)^2 = 0$, it is a single point; a negative right side gives no real locus; a difference equal to 0 gives a pair of straight lines.
• A family like $\dfrac{x^2}{p-k}+\dfrac{y^2}{q-k}=1$ is an ellipse when both denominators are positive and unequal, a hyperbola when they have opposite signs.