Consider the graph of the curve the square root of x the sq

Consider the graph of the curve the square root of x + the square root of y = a. Show that for all tangent lines to this graph the sum of the x and y intercepts is the same. Hint: let (h,k) be any point of the graph. find the equation of the tangent at (h,k) and find the intercepts and add them. Show that what you get is free of h and k so it would be the same for all tangents.

Solution

square root of x + the square root of y = a

h+k = a

h+2(hk)+ k = a^2

dy/dx= -(k/h)

equation of tangent at h,k

(y-k) = -(k/h) (x-h)

y intercept = (hk) + h

x intercept = (hk) + h

sum= h+2(hk)+ k = a^2

therefore constant