SOLUTION: Consider the curve y= 2/x with x>0. The normal to the curve at point P intersects the y axis at (0,-3). Find the point P? I understand the gradient of the normal is -2/x^2 and the

Algebra ->  Linear-equations -> SOLUTION: Consider the curve y= 2/x with x>0. The normal to the curve at point P intersects the y axis at (0,-3). Find the point P? I understand the gradient of the normal is -2/x^2 and the      Log On


   

Question 364660: Consider the curve y= 2/x with x>0. The normal to the curve at point P intersects the y axis at (0,-3). Find the point P?
I understand the gradient of the normal is -2/x^2 and the equation using (0,-3)
making 2/x = -2/x^2-3 so -2/x^2-2/x-3 but can't think of where else to go

Note: At somepoint in the in solving this problem you may be required to solve a non trivial possibly high degree polynomial equation. in that case trial and error may be required but note that the answer should be a possitive integer smaller than 5
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
You can get the value of the slope of the tangent line for the curve by taking the derivative.
m%5Bt%5D=-2%2Fx%5E2
The tangent and the normal to the curve are perpendicular to each other.
PErpendicular lines have slopes that are negative reciprocals.
m%5Bt%5D%2Am%5Bn%5D=-1
m%5Bn%5D=-1%2Fm%5Bt%5D
m%5Bn%5D=x%5E2%2F2
Now you have the slope of the normal line and you have one point (0,-3).
Use the point slope form of a line,y-y%5Bp%5D=m%28x-x%5Bp%5D%29
y-%28-3%29=%28x%5E2%2F2%29%28x-0%29
y%2B3=x%5E3%2F2
y=x%5E3%2F2-3
But you also know that,
y=2%2Fx
Substitute,
2%2Fx=x%5E3%2F2-3
Multiply both sides by 2x,
4=x%5E4-6x
x%5E4-6x-4=0
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graph%28300%2C300%2C-2%2C5%2C-2%2C3%2Cx%5E4-6x-4%29
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That curve has a zero at x=2
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P:(2,1)