Find a function with given condition

[gruba]

Homework Statement

Find a curve that passes through point [itex]A(2,0)[/itex] such that the triangle which is defined with a tangent at arbitrary point [itex]M[/itex], axis [itex]Oy[/itex] and secant [itex]\overline{OM}[/itex] is isosceles. [itex]\overline{OM}[/itex] is the base side of a triangle.

2. The attempt at a solution
Function passes through point [itex]A(2,0)\Rightarrow y=k(x-2),k\neq 0.[/itex]
There are two cases, for [itex]k<0[/itex] and for [itex]k>0[/itex].

What is the relation between points [itex]A[/itex] and [itex]M[/itex] because [itex]M[/itex] is not defined?

In which quadrant the point [itex]M[/itex] should be?

 

[pasmith]

M is an arbitrary point in the curve; its coordinates are (x, f(x)).

I think the assertion is that the distance along the tangent from M to the y-axis is equal to the distance from the origin to the intersection of the tangent and the y-axis. That allows you to write down a first-order differential equation which f must satisfy, and your initial condition comes from the constraint that A lies on the curve (ie f(2) = 0).