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Kissing

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Target Six

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Cocked Hat

Age 16 to 18 Challenge Level:

Expand the square the left hand side of the expression and rearrange the expression to give a quadratic equation in $y$. Now solve the quadratic equation. This will give two solutions (as quadratic equations generally do) and each will give you the equation of a branch of the graph. Sketch the graph by considering where real values of $y$ exist, where the graph crosses the axes and what symmetries it has. You can use graphing software but it is much more of a challenge to sketch the graph without.

Don't let this defeat you. It is not nearly as hard as it looks but you have to do a little algebra. If you wish you can choose a particular value for $a$, say $a=1$, and investigate one member of the family of graphs but then you will have to go back and see what happens for different values of $a$. You can use graphing software but it is much more of a challenge to sketch the graph without.