Mathematics CHORDS OF ELLIPSE

Chords of Ellipse- Equation of Chord of an Ellipse :

Equation of a chord of an ellipse joining two points `P(alpha)` and `Q(beta)` on it is equal to

`x/a cos ((alpha+beta)/2)+y/b sin ((alpha+beta)/2)= cos ((alpha-beta)/2)`

(use formula of line joining points `P(a cos alpha, b sin alpha)` and `Q(a cos beta, b sin beta))`

If this particular chord passes through `(d, 0)` then we have

`d/a cos ((alpha+beta)/2) =cos((alpha-beta)/2) ; (cos ((alpha+beta)/2))/(cos ((alpha-beta)/2))=a/d`

Using componendo and dividendo rule

`(cos ((alpha+beta)/2)-cos ((alpha-beta)/2))/(cos ((alpha-beta)/2)+cos ((alpha-beta)/2))=(a-d)/(a+d)`

or `-(2 sin alpha //2 sin beta//2)/(2 cos alpha//2 cos beta//2)=(a-d)/(a+d)`

i.e., `tan (alpha/2) tan (beta/2) =(e-1)/(e+1)`

Chords of Ellipse- Chord of Contact :

Pair of tangents drawn from outside point `P (x_1, y_1)` to the ellipse which meet it at `A` and `B`. Now line joining `A` and `B` is called the chord of contact of point `P (x_1, y_1)` w.r.t. the ellipse.

The equation of chord of contact is

`(x x_1)/(a^2)+(yy_1)/(b^2) =1`

Chords of Ellipse- Chord With a Given Middle Point :

Here chord `AB` is shown in the figure whose mid point is `P(x _1, y_1)`.

Then equation of this chord `AB` is `T = S_1`

Here `S_1 equiv (x_1^2)/(a^2)+(y_1^2)/(b^2)-1`

`T equiv (x x_1)/(a^2)+(yy_1)/(b^2)-1`

Diameter :

The locus of the middle points of a system of parallel chords with slope `'m'` of an ellipse is a straight line passing through the centre of the ellipse, called its diameter and has the equation

`y=-(b^2)/(a^2 m) x`.

Chord of contact, pair of tangents, chord with a given middle point, pole & polar are to be interpreted as same as they are in parabola.