Let 𝑃 be a point on the circle 𝑆 with both coordinates being positive. Let the tangent to 𝑆 at 𝑃 intersect the coordinate axes at the points 𝑀 and 𝑁. Then, the mid-point of the line segment 𝑀𝑁 must lie on the curve

A

$$(x+y)^{2}=3xy$$

B

$$x^{\frac{2}{3}} + x^{\frac{2}{3}} = 2^{\frac{4}{3}}$$

C

$$x^{2}+y^{2} = 2xy$$

D

$$x^{2}+y^{2} = x^{2}y^{2}$$