F7 coordinate geometry (short)

Let m be the slope of the tangents, i.e. m has 2 possible values, then the equation of the tangents are given by:

y - β = m(x - α)

y = mx + (β - mα)

Using the given info:

(β - mα)^2 = a^2 m^2 + b^2

β^2 + m^2 α^2 - 2αβm = a^2 m^2 + b^2

(a^2 - α^2) m^2 + 2αβm + (b^2 - β^2) = 0

So the slopes satisfy this quadratic equation and if they are perp. then their product will be -1, i.e.

(b^2 - β^2)/(a^2 - α^2) = - 1

b^2 - β^2 = α^2 - a^2

α^2 + β^2 = a^2 + b^2