Skip over navigation
Guide and features
Guide and features
Science, Technology, Engineering and Mathematics
Featured Early Years Foundation Stage; US Kindergarten
Featured UK Key Stage 1&2; US Grades 1-5
Featured UK Key Stage 3-5; US Grades 6-12
Featured UK Key Stage 1, US Grade 1 & 2
Featured UK Key Stage 2; US Grade 3-5
Featured UK Key Stages 3 & 4; US Grade 6-10
Featured UK Key Stage 4 & 5; US Grade 11 & 12
Why do this problem?
is a way to get into implicit functions using the familiar mathematics of quadratic equations. It would be a good discussion focus for the introduction of the topic of implicit functions. The third part allows students to make use of their calculus.
The first part of this question would make a good starter, the second and third parts would be well suited for individual calculation; they could be approached experimentally/numerically or using algebra.
Are you clear which parts are variables and which parts are constants?
How can we find $X$ in terms of $r$ directly?
For an exercise in complex numbers, you might try these extensions:
Which purely imaginary values of $r$ give purely imaginary or purely real values of $X$?
Which complex values of $r$ give real values of $X$?
Start off searching numerically for more real values of $r$ which give a real value of $X$. Who can find the smallest such value?
Mathematical reasoning & proof
Differentiation of parametric and implicit functions
Meet the team
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Register for our mailing list
Copyright © 1997 - 2014. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the
Millennium Mathematics Project