Ellipse

Terms Related to an Ellipse:

Fundamental Terms

Ellipse (Horizontal ellipse)

Conjugate Ellipse(Vertical ellipse)

(a)

Equation \(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a>b)\)

\(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a<b)\)

(b)

Graph

ellipse ellipse

(c)

Centre C (0, 0)

C (0, 0)

(d)

Vertices (±a, 0)

(0, ±b)

(e)

Length of major axis 2a

2b

(f)

Length of minor axis 2b

2a

(g)

Foci (±ae, 0)

(0, ±be)

(h)

Equation of directrices X = ±(a/e)

Y = ±(b/e)

(i)

Eccentricity \(e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}\)

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

(j)

Length of latusrectum 2b²/a

2a²/b

(k)

Ends of latusrectum (±ae, ±b²/a)

(±a²/b, ±be)

(l)

Parametric equations \(\left\{ \begin{align} & x=a\cos \alpha  \\ & y=b\sin \alpha  \\\end{align} \right\}\)

\(\left\{ \begin{align} & x=a\cos \alpha  \\ & y=b\sin \alpha  \\\end{align} \right\}\)

(m)

Parametric coordinates acosα, bsinα

acosα, bsinα

(n)

Focal distance or radii |SP| = (a – ex₁) and |S’P| = (a + ex₁)

|SP|=(b – ey₁) and |S’P| = (b + ey₁)

(o)

Sum of local radii |SP| + |S’P| 2a

2b

(p)

Distance between foci 2ae

2be