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- The standard equation of ellipse with reference to its principal axis along the coordinate axis is given by x
^{2}/a^{2 }+ y^{2}/b^{2}= 1 - In the standard equation, a >b and b
^{2}= a^{2}(1-e^{2}) Hence, the relation between a and b is a^{2}â€“ b^{2}= a^{2}e^{2}, where â€˜eâ€™ is the eccentricity and 0 < e < 1. - The foci of the ellipse are S(ae, 0) and Sâ€™ = (-ae, 0)
- Equations of the directrices are given by x = a/e and x = -a/e
- The coordinates of vertices are Aâ€™ = (-a, 0) and A = (a,0)
- The lengths of the major and minor axis are 2a and 2b respectively.
- The length of latus rectum is 2b
^{2}/a = 2a(1-e^{2}) - The sum of the focal distances of any pint on the ellipse is equal to the major axis. As a result, the distance of focus from the extremity of a minor axis is equal to semi major axis.
- If a question does not mention the relation between a and b then by convention a is assumed to be greater than b i.e. a > b.
- The point P(x
_{1}, y_{1}) lies outside, inside or on the ellipse according as x_{1}^{2}/a^{2}+ y_{1}^{2}/b^{2 }â€“ 1>< or = 0. - In parametric form, the equations x = a cos Î¸ and y = b sin Î¸ together represent the ellipse.
- The line y = mx + c meets the ellipse x
^{2}/a^{2}+ y^{2}/b^{2}= 1 in either two real, coincident or imaginary points according to whether c^{2}is < = or > a^{2}m^{2 }+ b^{2} - The equation y = mx + c is a tangent to the ellipse if c
^{2}= a^{2}m^{2}+ b^{2} - The equation of the chord of ellipse that joins two points with eccentric angles Î± and Î² is given byx/acos (Î± + Î²)/2 + y/b sin (Î± + Î²)/2 = cos (Î± - Î²)/2
- The equation of tangent to the ellipse at the point (x
_{1}, y_{1}) is given byxx_{1}/a^{2}+ yy_{1}/b^{2}= 1 - In parametric form, (xcosÎ¸) /a + (ysinÎ¸/b) is the tangent to the ellipse at the point (a cos Î¸ a, b sin Î¸)
- Equation of normal

1. Equation of normal at the point (x_{1},y_{1}) is

a^{2}x/x_{1}â€“ b^{2}y/y_{1}= a^{2}- b^{2}= a^{2}e^{2}

2.Equation of normal at the point (a cos Î¸ a, b sin Î¸) is ax secÎ¸ â€“ by cosec Î¸ = (a^{2}-b^{2})

3. Equation of normal in terms of its slope â€˜mâ€™ is

y = mx â€“ [(a^{2}-b^{2})m /âˆša^{2}+b^{2}m^{2}] - The equation of director circle is x
^{2}+ y^{2}= a^{2}+ b^{2 } - The portion of the tangent to an ellipse between the point of contact and the directrix subtends a right angle at the corresponding focus.

The perpendiculars from the center upon all chords which join the ends of any particular diameters of the ellipse are of constant length.

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