Equation of Ellipse in Standard Form

Equation of Ellipse in Standard Form

The equation of ellipse in standard form referred to its principal axes along the coordinate axes is

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

where a > b & \(b^2\) = \(a^2(1 – e^2)\) \(\implies\) \(a^2\) – \(b^2\) = \(a^2e^2\).

where e = eccentricity (0 < e < 1)

Foci : S = (ae, 0) & S’ = (-ae, 0)

Vertices : A’ = (-a, 0) and  A’ = (a, 0)

ellipse

(a) Equation of directrix of Ellipse

x = \(a\over e\)  and  x = \(-a\over e\)

(b) Major axis of Ellipse

The line segment A’A in which the foci S’ & S lie is of length 2a & is called the major axis (a > b) of the ellipse. The Point of intersection of major axis with directrix is called the foot of the directrix(z).

(c) Minor axis of Ellipse :

The y-axis intersects the ellipse in the points B’ = (0,-b) & B = (0,b). The line segment B’B of length 2b (b < a) is called the minor axis of the ellipse.

Both the axes minor and major together are called Principal Axes of the ellipse.

(d)  Double ordinate of Ellipse :

A chord perpendicular to major axis is called double ordinate of ellipse.

(e)  Latus Rectum of Ellipse :

The focal chord perpendicular to major axis is called the latus rectum of ellipse.

(i)  Length of latus rectum(LL’) = \(2b^2\over a\) = \({(minor axis)}^2\over {major axis}\) = 2a(1 – \(e^2\))

(ii) Equation of latus rectum : x = \(\pm\)ae

(iii)  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)  Eccentricity of Ellipse :

e = \(\sqrt{1 – {b^2\over a^2}}\)

Example : Find the equation of ellipse in standard form having center at (1, 2), one focus at (6, 2) and passing through the point (4, 6).

Solution : With center at (1, 2), the equation of the ellipse is \((x – 1)^2\over a^2\) + \((y – 2)^2\over b^2\) = 1. It passes through the point (4, 6)

\(\implies\) \(9\over a^2\) + \(16\over b^2\) = 1 …..(i)

Distance between focus and center = (6 – 1) = 5 = ae

\(\implies\) \(b^2\) = \(a^2\) – \(a^2e^2\) = \(a^2\) – 25 …..(ii)

Solving (i) and (ii)

we get \(a^2\) = 45 and \(b^2\) = 20

Hence, the equation of the ellipse is \((x – 1)^2\over 45\) + \((y – 2)^2\over 20\) = 1

Hope you learnt what is the equation of ellipse in standard form and its basic concepts, learn more concepts of ellipse and practice more questions to get ahead in the competition. Good luck!

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