Basics of Ellipse X=Acosθ and Y=Bsinθ

Introduction

The locus of a point in a plane which moves such that the ratio of its distance from a fixed point (focus) to a fixed line (directrix) is a constant less than one (eccentricity) is called an ellipse. Standard Equation

The standard equation of the ellipse is given by x2a2+y2b2 Co-ordinates of F (ae,0) Co-ordinates of F' (−ae,0) Equation of directrix corresponding to F is x=ae Equation of directrix corresponding to F is x=−ae Parametric form x=acosθ and y=bsinθ Latusrectum

LL′ is a latus rectum perpendicular to major axis and passing through one of the focii. There are two latus rectums passing through the two focii. ∣LL′∣=2b2a Focal property of ellipse The sum of the focal distance of any point on the ellipse is equal to the length of the major axis. ∣PF∣+∣PF′∣=2a

Auxiliary circle The circle of AA' as a diameter is called the auxiliary circle of the ellipse Tangent

Condition on a line to be a tangent Condition for line xcosα+ysinα=p to be a tangent p2=a2cos2α+b2sin2α. Tangent in terms of point of contact

Given the ellipse to be x2a2+y2b2=1 tangent at the point (x1,y1) on it is given by xx1a2+yy1b2=1

Tangent in terms of the slope Let m be the slope of the tangent, then the equation of tangent is y=mx±a2m2+b2−−−−−−−−−√ Tangent in terms of the parameter Let any point on ellipse be (a cos θ,b sin θ) Equation of tangent in parametric form: xcosθa+ysinθb=1 Normal

Equation of normal in terms of point of contact

Equation of ellipse: x2a2+y2b2=1 In the cartesian form equation of normal at point (x1,y1) is y−y1= y1a2x1b2(x−x1),x1≠0 Normal in terms of the slope Let m be the slope of the normal then the equation of the normal is y=mx±(a2−b2)ma2+b2m2−−−−−−−−−√ Equation of normal in terms of parameter In parametric form equation at the normal at (acosθ,bsinθ) is xasinθ−ybcosθ=(a2−b2)sinθcosθ Chord of an Ellipse

Equation of chord in terms of point of contact

Equation of chord having ends (x1,y1) & (x2,y2) is (x−x1)(x1+x2)a2+(y−y1)(y1+y2)b2=0 Equation of chord in terms of parameter Equation of chords with end points as (acosθ1,bcos θ1) and (acosθ2,bcos θ2) xacos(θ1+θ2)2+ybsin(θ1+θ2)2=cos(θ 1−θ2)2 Equation of chord in terms of its mid-point

Equation of chord in terms of its mid-point (x1,y1) is given by xx1a2+yy1b2=x21a2+y21b2 Director Circle

Director circle The locus of the point of intersection of the tangents to the ellipse which meet at right angle, is a circle. This circle is called the "director circle" of the ellipse. Equation of the director circle of the ellipse is x2+y2=a2+b2 Chord of contant

Equation of chord of contact

Equation of chord of contact of a point (x1,y1) is given as xx1a2+yy1b2=1 Polar of a point Polar of a point

xx1a2+yy1b2=1 is the equation of polar, where (x1,y1) is the pole

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is . This can be explained as follows. If we let an independent parameter increase from 0 to 2π, and let

and

then plotting x and y values for all angles of θ results in an ellipse. Note that is the eccentric anomaly and is not the angle traced out by a point on the ellipse (see below). For example, at θ = 0, x = a, y = 0 and at θ = π/2, y = b, x = 0. This can be seen as follows. Squaring both equations gives:

And

Dividing these two equations by a2 and b2 respectively gives:

And

Adding these two equations together gives:

Applying the Pythagorean identity to the right hand side gives:

This means any noncircular ellipse is a compressed (or stretched) circle. If a circle is treated like an ellipse, then the area of the ellipse would be proportional to the length of either axis (i.e. doubling the length of an axis in a circular ellipse would create an ellipse with double the area of the original circle). Focus The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

The sum of the distances from any point P = P(x,y) on the ellipse to those two foci is constant and equal to the major axis (proof):

. Eccentricity The eccentricity of the ellipse (commonly denoted as either e or ) is

(where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening factor

Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections. Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.