Mathematics CHORD OF HYPERBOLA

Chord of Hyperbola :

Chord joining two points with eccentric angles `alpha` and `beta` is given by

`x/a cos frac(alpha-beta)(2)-y/b sin frac(alpha+beta)(2)= cos frac(alpha+beta)(2)`.................(1)

if `(1)` passes through `(d, 0)` then

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

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

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

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

Chord of Contact :

If the tangents from a point `P(x_1, y_1)` to the hyperbola `x^2/a^2-y^2/b^2=1` touch the hyperbola at `Q` and `R`, then the
equation of the chord of contact `QR` is

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

Equation of the Chord Bisected at a Given Point

The equation of the chord of the hyperbola `x^2/a^2-y^2/b^2=1` bisected at the point `P (x_1 ,y_1)` is

`(x x_1)/a^2-(yy_1)/b^2=1=x_1^2/a^2-y_1^2/b^2-1`

or `T=S_1` where `Y=(x x_1)/a^2-(yy_1)/b^2=1` and `S_1=x_1^2/a^2-y_1^2/b^2-1`