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Math Challenge for Enthusiasts

The document describes a problem where a point A(a,b) is chosen in the (x,y)-plane. As the point P varies along the curve y=x^3, the midpoint M of the line segment between A and P traces out another cubic curve. The problem asks to (a) find the equation for the cubic traced out by M in terms of a and b, and (b) describe the points A(a,b) where the traced cubic intersects y=x^3 in 1-2 points.

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ElevenPlus Parents
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47 views1 page

Math Challenge for Enthusiasts

The document describes a problem where a point A(a,b) is chosen in the (x,y)-plane. As the point P varies along the curve y=x^3, the midpoint M of the line segment between A and P traces out another cubic curve. The problem asks to (a) find the equation for the cubic traced out by M in terms of a and b, and (b) describe the points A(a,b) where the traced cubic intersects y=x^3 in 1-2 points.

Uploaded by

ElevenPlus Parents
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Problem of the Month

Problem 6: March 2020

A point A(a, b) in chosen in the (x, y)-plane. For a point P that lies on the graph
of y = x3 , let M be the midpoint of AP . As the point P varies, so does the
midpoint M of line segment AP . In fact, as P varies, the point M traces out the
graph of another cubic equation.
(a) Find an equation for the cubic traced out by M . The coefficients of this
cubic should depend on a and b.
(b) Describe all points A(a, b) in the plane for which the traced-out cubic
intersects y = x3 in at least one but at most two distinct points.

Webpage: cemc.uwaterloo.ca/resources/potm.php Email: potm@uwaterloo.ca

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