Let l be any tangent line to the curve [tex]\sqrt{x}+\sqrt{y}=\sqrt{c}[/te x]. show that the sum of the x- and y-intercepts of l is c.

The question wants  c, so let us isolate c.if  , then squaring both sides of the equation (use foil or whatever method to expand the right side): then for any point  that is on the curve,  will be equal to===to find the equation for this tangent line, we need to do implicit differentiation to slope of the tangent line at a point .assume that c is a constant.differentiate both sides with respect to x.this tells us that for any point  on the given curve, there is a tangent line to that curve with a slope of  which uses the coordinate of point  .  the equation of the tangent line, in point-slope form,======the x-intercept of the tangent line is x-coordinate of where the graph of the tangent line crosses the x-axis. this occurs at y=0. let (a,0) be the coords of the x-intercept of the tangent line.leave  and  alone. those are based on coordinates on the curve. the x- and y- without subscripts in the equation of the tangent line are based on coordinates on the tangent line.the x-coordinate of the x-intercept is therefore======the y-intercept of the tangent line  is y-coordinate of where  the graph of the tangent line  crosses the y-axis.  this occurs at x=0.  let (0,b) be the coords of the  y-intercept of the tangent line.the y-coordinate of the y-intercept is therefore======sum of the x- and y-intercepts is c: consider the following: now consider  .if we square and square root  and  , then we can put them inside the radical. (we can assume they are always positive because in the original equation, they are under square roots.)