Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Cocked Hat

The implicit equation

$$(x^2 + 2a y -a^2)^2 = y^2(a^2 - x^2)$$

represents a family of functions the members of the family corresponding to different values of $a$.

Re-arrange the equation into the form $y= f(x)$ and draw some graphs belonging to the family for different values of $a$. What is the domain of the function $f(x)$?

To investigate these graphs you might try http://www.desmos.com.## You may also like

### Kissing

### Good Approximations

### Target Six

Or search by topic

Age 16 to 18

Challenge Level

The implicit equation

$$(x^2 + 2a y -a^2)^2 = y^2(a^2 - x^2)$$

represents a family of functions the members of the family corresponding to different values of $a$.

Re-arrange the equation into the form $y= f(x)$ and draw some graphs belonging to the family for different values of $a$. What is the domain of the function $f(x)$?

To investigate these graphs you might try http://www.desmos.com.

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?

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

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