Find the sum of the x and y intercepts of any tangent line to the curve sqrtx + sqrty = sqrt7

I don't understand, please explain in several steps.
Thank you!

x^(1/2) + y^(1/2) = 7^(1/2)
(1/2)x^(-1/2) + (1/2)y^(-1/2)*dy/dx = 0
dy/dy = -(1/2)x^(-1/2)/(1/2)y^(-1/2) = -√(y/x)
dy/dy = -1 when y = x and 2x^(1/2) = 7^(1/2)
x^(1/2) = 7^(1/2)/[4^(1/2)] = [7/4]^(1/2)
Symmetrical mid point x = y = 7/4 on a line with slope -1
Intercepts must be double this, so both 7/2
Intercepts on this line, (and any other tangent line), sum to 7.

With hindsight when y = 0, x^(1/2) = 7^(1/2), intercepts sum to 7.
Similar result with x = 0.

Regards - Ian