Algebra problem 1 by Dhaval Furia

(a^2 + b^2) * (x^2 + y^2) = 25 * 169

a^2 x^2 + a^2 y^2 + b^2 x^2 + b^2 y^2 = 5^2 * 13^2

(a^2 x^2 + b^2 y^2) + (a^2 y^2 + b^2 x^2) = 65^2

((ax+by)^2 - 2abxy) + ((ay-bx)^2 + 2abxy) = 65^2

65^2 - 2abxy + k^2 + 2abxy = 65^2

k^2 = 65^2 - 65^2 = 0

k = 0

Let $a=5\cos α, b=5\sin α,x=13\cos β,y=13\sin β$ . Then

$ax+by=65\cos(α-β)=65\implies α=β$ .

$k=65\sin (β-α)=\boxed 0$ .