Building blocks

Circles

In the diagram below, $P$ is a point on a circle of radius $1$ and $\theta$ is the angle between $OP$ and the positive $x$-axis.

We have drawn in the diameter through $P.$

• $M$ is the point where the diameter through $P$ meets the line $x=1$
• $N$ is the point where the diameter through $P$ meets the line $y=1$

Using the functions from the triangles problem, work out the coordinates of points $P$, $M$ and $N.$

As $\theta$ varies, how will $M$ and $N$ move? What happens when $\theta$ isn’t acute?

• When will $M$ and $N$ be moving quickly?
• Do $M$ and $N$ always move smoothly?
• When will their $x$- or $y$-coordinates change sign?

In the next diagram, we have drawn the tangent to the circle at $P$ and marked the points where the tangent at $P$ meets the axes.

What are the coordinates of $K$ and $L$? How will $K$ and $L$ move along the axes as $\theta$ varies?