.i have this project due and ive been stumped on it for awhile
It goes like this The parabola y=x^2 has a circle with a radius 1 centered on the y axis and is tangent to the parabola.
1. let (0,b) be the the center of the circle on the y axis that is tangent to the parabola. state the equation of the circle.
answer: x^2-(y-b)^2=1 Should be x^2+(y-b)^2=1
2. Find the derivative of the parabola y=x^2 at any x.
answer: f'(x)= 2x
3. Find the derivative of the circle at any arbitrary point on the graph. use implicit differentiation.
answer dy/dx= x/(y-b) dy/dx= -x/(y-b)
This is where i need help!!
4. since the derivative in step 2 and step 3 are equal where they are tangent, set the two derivatives equal and get an equation involving y and b. So put those two derivatives equal to each other: 2x = -x/(y-b). Notice how the x's cancel, to give you an equation involving only y and b.
5. Since the two graphs intersect, solve for one variable in one equation and substitute into the other one to get a second equation involving y and b.(substitution method) If y=x^2 and x^2+(y-b)^2=1 then y+(y-b)^2=1.
6. Use the two equations from steps 4 and 5 to solve for b. Now you do the rest!
7. Now write the equation of the circle.
8. Find the points of intersection of the circle and parabola.