Where does the normal line to the parabola #y=x-x^2# at the point (0,1) intersect the parabola a second time?

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Answer 1 · 2017-01-17T19:34:24

It intersects only once. See below for proof.

Differentiate using the power rule.

#y' = 1 - 2x#

Determine the slope of the tangent:

#m_"tangent" = 1 - 2(0) = 1#

The normal line is perpendicular to the tangent.

#m_"normal" = -1/m_"tangent" = -1#

Determine the equation now using the point-slope form of the equation of a line, #y - y_1 = m(x - x_1)#.

#y - 1 = -1(x - 0)#

#y - 1 = -x#

#y = 1 - x#

If we want to see when the line intersects the parabolas, write a system of equations and solve.

#{(y = 1 - x), (y = x - x^2):}#

Solve through substitution.

#1 - x = x - x^2#

#x^2 - 2x + 1 = 0#

#(x - 1)(x - 1) = 0#

#x = 1#

This shows that the normal line at #x = 0# intersects the parabola only once.

Hopefully this helps!