Different Types of Ellipse Equations and Graph

Here, you will learn different types of ellipse and their basic definitions with their graphs.

Let’s begin –

Different Types of Ellipse

(a)  First type of Ellipse is

\(x^2\over a^2\) + \(y^2\over b^2\) = 1, where a > b

ellipse graph

(a)  AA’ = Major axis = 2a

(b)  BB’ = Minor axis = 2b

(c)  Vertices = (\(\pm a\), 0)

(d)  Latus rectum LL’ = L1L1′ = \(2a^2\over b\), equation y = \(\pm\)ae

(e)  Ends of latus rectum are : L(ae, \(b^2\over a\)), L'(ae, -\(b^2\over a\)), L1(-ae, \(b^2\over a\)), L1′(-ae, -\(b^2\over a\))

(f)  Equation of directrix y = \(\pm a\over e )

(g)  Eccentricity : e = \(\sqrt{1 – {b^2\over a^2}}\)

(h)  Foci : S = (ae, 0) & S’ = (-ae, 0)

(b)  Second type of Ellipse is

\(x^2\over a^2\) + \(y^2\over b^2\) = 1 (a < b)

ellipse1 graph

(a)  AA’ = Minor axis = 2a

(b)  BB’ = Major axis = 2b

(c)  Vertices = (0, \(\pm b\))

(d)  Latus rectum LL’ = L1L1′ = \(2a^2\over b\), equation y = \(\pm\)be

(e)  Ends of latus rectum are : L(\(a^2\over b\), be), L'(-\(a^2\over b\), be), L1(\(a^2\over b\), -be), L1′(-\(a^2\over b\), -be)

(f)  Equation of directrix y = \(\pm b\over e\)

(g)  Eccentricity : e = \(\sqrt{1 – {a^2\over b^2}}\)

(h)  Foci : S = (0, be) & S’ = (0, -be)

Line and an ellipse

The line y = mx + c meets the ellipse \({x_1}^2\over a^2\) + \({y_1}^2\over b^2\) = 1 in two real, coincident, or imaginary according as \(c^2\) is < = or > \(a^2m^2 + b^2\).

Hence y = mx + c is tangent to the ellipse \({x_1}^2\over a^2\) + \({y_1}^2\over b^2\) = 1 if \(c^2\) = \(a^2m^2 + b^2\)


Related Questions

The foci of an ellipse are \((\pm 2, 0)\) and its eccentricity is 1/2, find its equation.

Find the equation of the ellipse whose axes are along the coordinate axes, vertices are \((0, \pm 10)\) and eccentricity e = 4/5.

If the foci of a hyperbola are foci of the ellipse \(x^2\over 25\) + \(y^2\over 9\) = 1. If the eccentricity of the hyperbola be 2, then its equation is :

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