CirclesIntroduction

Riya is setting the table for dinner. She puts a round steel thali down and, just for fun, lays her ruler across it. Sometimes the edge of the ruler crosses the rim of the thali twice. Sometimes it stays clear of the thali altogether. And once, if she is very careful, it just grazes the rim at a single point — the way a ball resting on a table touches it at one tiny spot.

In Class IX you met the circle: all the points of a plane that are the same distance — the radius — from one fixed point, its centrecircumferencediameter. You also met its chord, arcs, segments and sectors. This chapter asks a new question: in how many different ways can a straight line and a circle meet?

Fig. 10.1 is live. The circle is Riya's thali and the line PQ is the edge of her ruler. Read the three missions, then drag the round handle to slide PQ up and down, letting go each time PQ is in place.

Each case left a line behind, and together they make the three pictures of Fig. 10.1. The dashed grey line (i) has no point in common with the circle: it is a non-intersecting line. The green line (ii) meets the circle at two points, A and B: it is a secant of the circle. The red line (iii) meets the circle at exactly one point, A: it is a tangent to the circle.

The readout gives a quick test. PQ was a tangent exactly when its distance OM from the centre was equal toless thangreater than the radius, and a secant when OM was less thangreater thanequal to the radius.

From now on the colours always mean the same thing: the circle is blue, tangents are red, secants and chords are green, and radii are purple.

Riya's nani lives in a village where the water still comes up from a well. A small wheel — a pulley — turns on an axle above the well. The rope runs over the top of the wheel and hangs straight down on both sides.

Fig. 10.2 — a pulley fitted over a well.

Think of the wheel as a circle and of each hanging part of the rope as a ray. Each one leaves the wheel at a single point and never cuts into it, so each is like a tangentsecantchord to the circle.

Exactly — there is no other position of a line with respect to a circle. In this chapter we ask when tangents exist, how many there can be, and prove two properties of tangents that the board exam asks about every year.

Speed Round: Miss, Cut or Touch?

You don't need a drawing to decide. Compare the distance of the line from the centre with the radius: if the distance is bigger, the line misses; if it is equal, the line touches; if it is smaller, the line cuts.

Get five right to move on.