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Homework Statement
andHomework Equations
[/B]I was working on a math project involving certain simulations and a quarter circle. The equation of the quarter circle I used is y = r - SQRT(r^2 - x^2). Where r is the radius of a circle and x is greater than 0.
The purpose of this is to model is to get a concave up, quarter-circle that runs through both the origin and a chosen point (x_final, y_final) where both x and y are again positive. This point must not necessarily be at the end of the quarter circle (i.e. when the gradient is vertical).
When I was running through my simulations I noticed that this model broke down if x_final was smaller than y_final. Hence the question is that is x_final greater than y_final a necessary condition of the curve.
The Attempt at a Solution
It makes intuitive sense to me that this condition must be true because if I imagine all the possible circles of different radii that rest with their bottom on the origin and x-axis, a point where x_final is smaller than y_final can only lie on the upper left quarter of the circle. It almost feels like x_final = y_final is the limit to this model. However, I can't think of a proper mathematical way to show this.