Statement-1: Two tangents drawn from any point on he hyperbola x2−y2=a2−b2 to the ellipse x2a2+y2b2 =1 make complementary angles with the axis of the ellipse.Statement-2: If two lines make complementary angles with the axis of x then the product of their slopes is 1.

If the angle in statement-2 are α and β such that α+β=π/2 then the product of the slopes is tan αtan⁡β = 1, and the statement-2 is true.In statement-1, any tangent to the ellipse isy=mx+a2m2+b2 which passes througha2−b2sec⁡θ,a2−b2tan⁡θ⇒a2−b2(tan⁡θ−msec⁡θ)2=a2m2+b2Product of the slopes =a2−b2tan2⁡θ−b2a2−b2sec2⁡θ−a2=a2sin2⁡θ−b2a2sin2⁡θ−b2=1So by statement-2, statement-1 is also true.