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Question 27358: We have had extreme trouble with this one. There are four of us and each of us comes up with a different polynomial to the fourth degree. Here it is: We have a circle (x-1)^2 + y^2 = 8 intersected by the parabola y = 1/4(x-1)^2 + 1. The question wants us to find the points of intersection between the parabola and the circle. We can see by the graph that the parabola intersects the circle at (-1,2) and (3,2) but we are having trouble with the actual equation. We know that we need to substitute and get this equation: (x-1)^2 + [1/4(x-1)^2 + 1]^2 = 8. This is where we are having troubles with simplifying the polynomial - we all get a different polynomial to the fourth degree. Can someone help? We would really appreciate a step-by-step to see where we are going wrong.
Thanks for your help
Melissa
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Solve the parabola equation for (x-1)^2 as follows:
Multiply through be "4" to get:
4y-4=(x-1)^2
Substitute that into the circle equation to get:
4y-4+y^2=8
y^2+4y-12=0
(y+6)(y-2)=0
y=-6 or y=2
Plug those values into the original circle equation one
at a time:
If y=-6 then (x-1)^2+36=8
Then (x-1)^2=-28
This has no Real number solution so forget it.
If y=2 then (x-1)^2+4=8
(x-1)^2=4
So, x-1=2 or x-1=-2
x=3 or x=-1
Solutions: (3,2) (-1,2)
Cheers,
stan H.
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