How many tangents can be drawn to a circle from a point (i) inside, (ii) on and (iii) outside the circle? Explain with figures.

1. 1 · Write what is given and what is to be found Given: a circle with centre O and a point P. To find: the number of tangents from P in three positions.
2. 2 · Draw the figure Three small circles: P inside; P on the circle with one tangent; P outside with two tangents PA and PB.
3. 3 · Write the definition A tangent meets the circle at exactly one point.
4. 4 · Case (i): P inside Every line through P cuts the circle at two points. Number of tangents = 0.
5. 5 · Case (ii): P on the circle Only one line through P touches the circle. Number of tangents = 1.
6. 6 · Case (iii): P outside Two lines through P just touch the circle. Number of tangents = 2.
7. 7 · Check and box the answer Write the result: inside 0, on 1, outside 2.

Scoring tip: Draw three separate small figures, one for each case, and label P in each. The figures carry a mark of their own.