I know this is a lot but i need somewhere to start.
Consider the curves in the first quadrant that have equationswhere 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 .
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 .