To find: The position of the point P in the given process and to check whether P have a limiting position.
The limiting position is .
Let the coordinate system and drop a perpendicular from P.
From the triangle the angle that ,
And from the triangle the angle that .
By using double-integral formula for tangents,
Simplifying above equation
Applying cross multiplication
As the altitude AM decreases in length, the point P will approach the x-axis, means y approaches to 0. So the limiting location of P exists and it must be one of the root of the equation. The point P can never be to the left of the altitude AM, so here x not equal to zero, so it must be implies .
Thus, the limiting position is .
To find: The equation of the curve and sketch the path traced out by P.
The P only traces out the part of the curve with .
Take the equation (1) from the part (a) is the equation of the curve traced out by P.
As approach infinity, approach , approach , x approach 1,
, y approach 1.
Thus, P only traces out the part of the curve with .
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