Here I describe four applet-based exercises that use variation to stress the following awareness: a tangent meets a circle at a single point <=> the simultaneous solution of their equations will result in a quadratic equation with a repeated root.
The first exercise, in which the equation of the line is varied but the circle remains invariant, is designed to highlight the connection between the number of solutions to the simultaneous equation and number of times the line meets the circle. Questions I will ask alongside this include: [putting the line in any position] If I solve these two equations simultaneously, can you predict how many solutions there will be? [then, moving the line] And now? And now?
The second exercise, in which the centre of the circle is varied but the tangent remains invariant, is designed to draw attention to the (moving) point of tangency, and the relationship between this and the solution to the simultaneous equation, which will be of the form 5(x-a)^2 = 0.
What matters is not only the tasks, but how they are used. Alongside each exercise, I will invite learners to predict before acting: What do you expect will happen when you solve these simultaneous equations?
The third exercise, in which the equations of the circles are varied but the point of tangency remains invariant, is designed to draw attention to the fact that the solution to the simultaneous equations will be of the form k(x-2)^2 = 0.
The fourth exercise, in which the tangent 'rotates' around an invariant circle, is designed to draw attention to the infinite tangents that one circle can have. This exercise might not be necessary, but I am aware that we have only worked with tangents with gradient 2 at this point. I wanted to vary this, to avoid automation when substituting the equation of the line into the circle equation.
Underlying each of these exercise is a subsidiary awareness, that the radius and tangent have perpendicular gradients. I have not stressed this because (a) the learners I am working with have a good grasp of this, and (b) attending to this might reduce the chance of learners attending to the key awareness, the 'algebraic representation' of tangency.