Question

If the tangent at (α, β) of x2/a2 - y2/b2 =1 meets the auxiliary circle at (x1,y1) and (x2,y2), then show that β is harmonic between y1 and y2 .

Answer 1

The tangent at (α, β) is

xα/a2 - yβ/b2= 1 ..(1)

and the auxillary circle is

x2 + y2 = a2 ....(2)

Substituting

Let y1 and y2 be the roots of this quadratic equation. Then

Therefore

Hence, β is harmonic between y1and y2.