Please verify my solution to this calc problem

[dnylander]

whitch of agnesi problem) its a curve defined as follows. let O be the origin, let T be the point (0,1), and let m be the line through T parallel to the x-axis. Let C be the circle centered at (0,0.5) with radius 0.5. For any point P on the circle besides O and T, we draw a ray from O through P. let this ray intersex m at point X. we then draw the altitude from X to the line through P parallel to the x-axis. The foot of this altitude, point A, is on the witch curve. When we trace out the resulting points A for all possible P, and include point T, we get the witch curve. Find the Parameterization for it and a function f such that the witch curve is the graph of the function y = f(x)

My answer:
a= radius
x= 2acos(t)
y= a[1-cos 2(t)]
y= 8a^3/x^2+4a^2

Thanks

 

[mighty2000]

for your post! The curve you are referring to is called the "Witch of Agnesi" curve, and it is a very interesting mathematical concept. The parameterization you have provided is correct, but there are a few things I would like to clarify.

First, the parameter "a" in your equations should actually be "r", which represents the radius of the circle. This is because the circle in this problem is centered at (0, 0.5) with a radius of 0.5, so the value "a" in your equations would actually be 0.5.

Second, the function you have provided is not the correct one for the curve. The correct function for the Witch of Agnesi curve is y = 2ar^2/(x^2+r^2). This function is derived from the parameterization you provided, but it takes into account the fact that the radius of the circle is not fixed at 0.5, but can vary.

So, in summary, the correct parameterization for the Witch of Agnesi curve is:
x = 2arctan(t)
y = 2ar^2/(1+t^2)

And the correct function for the curve is:
y = 2ar^2/(x^2 + r^2)

I hope this helps clarify any confusion and thank you for your contribution to the forum!