Coordinate Geometry

Hyperbola

Types of hyperbolas, eccentricity, foci, tangents, normals, and focal distances.

A hyperbola is the locus of points whose distances to two fixed foci have a constant difference in absolute value (2a), rather than a constant sum as in the ellipse. The curve has two disconnected branches opening away from each other along the transverse axis. Its eccentricity e is always greater than 1, and as |x| or |y| grows the branches approach a pair of asymptotes through the centre. The two standard orientations place the transverse axis along the x-axis (Type 1) or the y-axis (Type 2).

Difference of distances to the two foci is constant: 2a.

Types of Hyperbola

The two standard orientations have transverse axes along x and y respectively.

Type 1

A horizontal hyperbola opens left-and-right, with vertices at (±a, 0) on the x-axis and foci at (±ae, 0). The transverse axis (length 2a) lies along x; the conjugate axis (length 2b) lies along y and does not intersect the curve. The asymptotes are y = ±(b/a) x.

\frac{x^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1

Type 2

A vertical hyperbola opens up-and-down, with vertices at (0, ±b) on the y-axis and foci at (0, ±be). The transverse axis (length 2b) lies along y; the conjugate axis (length 2a) lies along x. The asymptotes are y = ±(b/a) x, the same as in Type 1.

\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1

Hyperbola	Type 1	Type 2
Center	(0,0)	(0,0)
Vertices	A(a,0) A’(a,0)	B(0,b) B’(0,-b)
Length of Axis	Transverse = 2a, Conjugate = 2b	Transverse = 2b, Conjugate = 2a
Equation of Axis	Transverse axis: y = 0, Conjugate axis: x = 0	Transverse axis: x = 0, Conjugate axis: y = 0
Distance Between Foci	2ea	2be

Relation between a,b and e (Type 1)

The identity b² = a²(e² − 1) relates the semi-axes to the eccentricity. (e² − 1) is positive because e > 1, so b² is positive as required. This is the hyperbolic analogue of the ellipse relation b² = a²(1 − e²), with the sign flipped.

b^{2}=a^{2}\left(e^{2}-1\right)

Relation between a,b and e (Type 2)

Type 2 swaps a and b, so the identity becomes a² = b²(e² − 1). Here b is the semi-transverse axis and a the semi-conjugate. The constraint e > 1 keeps the right side positive.

a^{2}=b^{2}\left(e^{2}-1\right)

Eccentricity (Type 1)

Solving b² = a²(e² − 1) for e gives e = √(a² + b²) / a, strictly greater than 1. Larger eccentricity spreads the asymptotes further from the transverse axis, so the branches diverge more sharply. The product ae is the centre-to-focus distance, mirroring the ellipse.

e=\frac{\sqrt{a^{2}+b^{2}}}{a}

Eccentricity (Type 2)

For Type 2 the semi-transverse axis is b, so the formula divides by b instead of a. The radicand is symmetric in a and b; only the denominator changes. The result is strictly greater than 1.

e=\frac{\sqrt{b^{2}+a^{2}}}{b}

Foci (Type 1)

The two foci sit on the transverse axis, symmetric about the centre, at distance ae from the origin. Since e > 1 and the vertices are at (±a, 0), each focus lies further out than the corresponding vertex, outside the curve on the side of the matching branch.

S=\left(\pm ae,0\right)

Foci (Type 2)

For Type 2 the foci lie on the y-axis at (0, ±be), symmetric about the centre. The distance from centre to focus is be, exceeding the semi-transverse axis b because e > 1. Each focus is outside the matching branch.

S=\left(0,\pm be\right)

Tangent to Hyperbola

The tangent forms mirror those of the ellipse, with the sign on the b² term flipped to match the hyperbola equation. The point form applies T = 0: x² → x x₁ and y² → y y₁. The contact point must satisfy the original equation; otherwise the line is the polar of an external point.

\left(i\right)\text{ }\text{ }\frac{\left(xx_{1}\right)}{a^{2}}-\frac{\left(yy_{1}\right)}{b^{2}}=1\text{ }\text{ }\left[at\text{ }P\left(x_{1},y_{1}\right)\right]

The slope form gives tangents of slope m; the radicand a²m² − b² must be non-negative, so tangents exist only for |m| ≥ b/a. The boundary case |m| = b/a corresponds to the asymptotes, tangents “at infinity”. Slopes between −b/a and b/a admit no tangent line.

\left(ii\right)\text{ }\text{ }y=mx\pm\sqrt{a^{2}m^{2}-b^{2}}\text{ }\text{ }\text{ }\text{ }\left[When\text{ }slope = m\right]

Normal to Hyperbola

The normal at (x₁, y₁) is perpendicular to the tangent and passes through the contact point. The formula is the ellipse normal with both signs flipped: the y term gains a plus and the right-hand side becomes a² + b². The form is undefined when x₁ or y₁ vanishes; at vertices the normal coincides with the transverse axis.

\frac{\left(a^{2}x\right)}{x_{1}}+\frac{b^{2}y}{y_{1}}=a^{2}+b^{2}

Focal Distance

A point P(x₁, y₁) on the hyperbola has two focal distances SP and S’P, one to each focus, with absolute difference 2a (the defining locus property). The sign of the difference depends on which branch P sits on. The closed forms below are linear in x₁, so the focal distances compute without square roots once e is known.

if\text{ }P=\left(x_{1},y_{1}\right)\text{ }then,

SP=\left|ex_{1}-a\right|

S'P=\left|ex_{1}-a\right|