Counting shared points sorts every line into one of three kinds

What to assume they know

A circle as the set of points at one fixed distance from a fixed centre, and the words centre, radius, chord, arc, segment and sector — all carried over from Class IX and assumed by name on the chapter's opening page
That the distance from a point to a line means the perpendicular distance
Pythagoras' theorem in a right triangle
That two points determine exactly one straight line

What they should be able to do

Sort a line drawn against a circle into one of exactly three kinds by counting the points the two share
Name the three kinds using the chapter's own vocabulary and say which count goes with which name
Explain why no fourth kind can exist, in terms of how many points of a circle a straight line can pass through
Compare the centre-to-line distance with the radius and predict the count of shared points before drawing anything
Compute the chord a line cuts, given the radius and the centre-to-line distance
Identify the tangent case as an exact equality and explain why "nearly tangent" is not a fourth case
Distinguish a chord from the secant that carries it, and a tangent line from the single point where it meets the circle
Point to a tangent in a physical arrangement such as a rope over a pulley

Where it usually goes wrong

"A tangent is a line that touches but does not cross." For a circle the two descriptions happen to agree, but "does not cross" is a statement about sides and is not what is being defined here. The definition on p. 144 is a count: one shared point. Students who carry the "does not cross" version forward meet trouble the moment they see any curve that is not a circle.
"A chord and a secant are the same thing." The chord is the segment between the two shared points and has ends; the secant is the whole line and does not. The distinction is what lets §10.2 talk about a chord shrinking to nothing while the line stays put.
"A line very close to the circle is almost a tangent." Either the centre-to-line distance equals the radius or it does not. There is no third verdict, and the chapter's later proofs would all collapse if there were — Theorem 10.1 uses the exactness of "only one shared point" as its whole hypothesis.
"Non-intersecting means parallel." There is nothing here for the line to be parallel to. It means the line and the circle have no point in common, and a non-intersecting line can point in any direction at all.
"Which case you get depends on how the line is tilted." Tilt the line about a fixed point and the case can change, but only because the tilt changed the centre-to-line distance. Slide the line without turning it and you can move it through all three cases with the direction never changing — this is exactly what Fig. 10.3 (ii) does on the next page.
"There are three kinds of line, so a line has a kind." A line has a kind only relative to a chosen circle. The same line is non-intersecting for one circle, a secant for another and a tangent for a third.

Questions to check understanding

Given a figure, name each drawn line as non-intersecting, secant or tangent
Fill-in-the-blank on the count of points a tangent shares with the circle, and on the name for a line sharing two
Given a radius and a centre-to-line distance, say how many points the line and the circle share
Given a radius and a chord length, find the centre-to-line distance, or the reverse
Construct, to instruction, one touching line and one cutting line, both parallel to a stated direction
Short-answer: explain why a straight line cannot meet a circle at three points

Examples worth working on the board

Inputs only. Values marked verified are worked out here on the chapter's printed data.

Fig. 10.1, the three positions (§10.1, p. 144). Read from the printed page: three circles in a row, each with the same line drawn as a straight arrow-headed segment labelled P at one end and Q at the other. In (i) the arrow runs down the left of the circle, clear of it. In (ii) it runs across the circle, entering near the top at a point marked A and leaving lower down at a point marked B. In (iii) it runs vertically down the left side and just grazes the circle at a point marked A. The three panels are labelled (i), (ii), (iii) beneath. The whole argument of this topic is that these are three counts — 0, 2, 1 — and not three shapes.
The count is the definition, so say the numbers out loud. Panel (i) shares nothing, panel (ii) shares two points, panel (iii) shares one. The chapter presents them in that order — none, two, one — which is not the numerical order; the explanation may prefer to walk 0 → 1 → 2 or 2 → 1 → 0, but should say that it is reordering.
Why there is no fourth case. Suppose a line met the circle at three distinct points. Two of them already fix the line, and the third would then have to sit on the same line and the same circle. But the perpendicular from the centre onto a line meets it once, and the points of the circle on that line sit at equal distances on either side of that foot — so at most two of them exist. This argument is added here. The chapter states plainly that no other position is possible and offers no reason; supplying one is the difference between a rule and an explanation.
The measurement that decides it. Let r be the radius and d the perpendicular distance from the centre to the line. Verified as algebra: a point of the line at distance t from the foot of that perpendicular sits at distance √(d² + t²) from the centre, so it lies on the circle exactly when t² = r² − d². That has two solutions when d < r, one when d = r, and none when d > r — reproducing the three cases and nothing else. This derivation is added here; the chapter never measures the distance from the centre to the line in §10.1.
A concrete run of the three cases. Take a circle of radius 5 cm — the radius the chapter itself keeps returning to in Example 3 (p. 150) and in Exercise 10.1 question 3 (p. 147). Verified: at d = 3 cm the line cuts a chord of 2√(25 − 9) = 8 cm; at d = 5 cm the chord has collapsed to length 0 and the line is a tangent; at d = 7 cm there is nothing to cut. The 5 cm and 8 cm pair is exactly Example 3's data read backwards, so the explanation can plant it here and collect it four topics later.
The pulley over the well (§10.1, p. 145, Fig. 10.2). Read from the printed page: a hatched horizontal beam at the top, a hook and bracket beneath it, a grooved wheel, and rope hanging down on both sides with a downward arrow on each strand. The chapter's point is that each hanging strand, extended as a ray, sits against the wheel's circle in the tangent position. Note: this is an illustration, not evidence — it shows where the third case turns up, not why it is one of only three.
Exercise 10.1 question 2, items (i) and (ii) (p. 147). Two fill-in-the-blank prompts: how many points a tangent shares with the circle, and what a line sharing two points is called. Verified from §10.1: one, and secant. These are the definitions of this topic tested directly.
Exercise 10.1 question 4 (p. 147). Set a circle down, then produce two lines that both run parallel to some stated direction, arranging one to touch the circle once and the other to cut across it. Hand this over as a construction task: it forces the student to realise that the direction of the line is fixed and only its distance from the centre is free, which is precisely the claim of section 6.

Figures to have open

Fig. 10.1's three panels as one redrawn strip, with the shared points marked as dots and a numeral 0, 2, 1 set under each. This is the chapter's own figure (p. 144) and must be redrawn rather than reproduced; the redraw should keep the arrowheads on the line, because they are what say the line has no ends.
A single circle with a line sliding across it under a fixed direction, shown step by step through the three cases, with a live readout of the centre-to-line distance and the chord length. Standard schematic, and entirely not in the book — the chapter draws nothing like it in §10.1.
The right triangle formed by the centre, the foot of the perpendicular, and one crossing point, labelled d, t, r. Standard schematic; not in the book.
Fig. 10.2's pulley (p. 145), or a simplified line drawing of a wheel with two rope strands hanging. The printed figure is a shaded pictorial drawing; a clean schematic serves the argument better and avoids reproducing the art.

Where this sits in the book

NCERT Mathematics, Textbook for Class X, Chapter 10 "Circles", §10.1 Introduction, pp. 144–145, with Fig. 10.1 (p. 144) and Fig. 10.2 (p. 145)
The footnote on p. 145 giving the Latin origin of the word tangent and naming Thomas Fineke and the year 1583
Exercise 10.1 questions 2 (i), 2 (ii) and 4, p. 147
The term point of contact, used here as a forward pointer, is defined in §10.2, p. 146
The chapter's summary, §10.4, p. 153, lists the meaning of a tangent as its first point