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Date Posted: 5/12/2008 12:48:14 AM  Status: Live
implicit differentiation
Course Textbook Chapter Problem
Calculus N/A N/A N/A
Question Details:
 
 
Suppose that a circle of radius r and center (0,3) is inscribed in the parabola y=x2. Find the points of tangency and the radius r. (Hint: What is the equation of the circle? How do the slopes of each curve compare at the points of tangency?)
 
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Date Posted: 5/12/2008 2:01:48 AM  Status: Live
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Response:
the equation of the circle is x2 + (y - 3)2 = r2
Usually the circle and the parabola have 4 points of intersection, but if they are tangent each other, only 2 points of tangency. By symmetry, these 2 points have same y.
Combine x2 + (y - 3)2 = r2 and y = x2, we have
y + (y - 3)2 = r2 ,
y2 - 5y + 9 - r2 = 0
when b2 - 4ac = 0, only one y
25 - 4(9 - r2) = 0, get r = √11/4
the one y = 5/2 = 2.5
x = ±√2.5
so
the points of tangency are (√2.5, 2.5) and (-√2.5, 2.5); r = √11/4



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