# Find a function with given condition

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1. Apr 22, 2016

### gruba

1. The problem statement, all variables and given/known data
Find a curve that passes through point $A(2,0)$ such that the triangle which is defined with a tangent at arbitrary point $M$, axis $Oy$ and secant $\overline{OM}$ is isosceles. $\overline{OM}$ is the base side of a triangle.

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

What is the relation between points $A$ and $M$ because $M$ is not defined?

In which quadrant the point $M$ should be?

2. Apr 22, 2016

### 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).