I know this is a lot but i need somewhere to start.

Consider the curves in the first quadrant that have equations where is a positive constant.

Different values of give different curves. The curves form a family, .

Let Let be the member of the family that goes through P.

A. Let be the equation of . Find .

B. Find the slope at of the tangent to .

slope =

C. A curve is perpendicular to at . What is the slope of the tangent to at the point ? slope =

D. Give a formula for the slope at of the member of that goes through . The formula should not involve or .

E. A curve which at each of its points is perpendicular to the member of the family that goes through that point is called an orthogonal trajectory to . Each orthogonal trajectory to satisfies the differential equation

where is the answer to part D.

Find a function such that is the equation of the orthogonal trajectory to that passes through the point .