Exercise 10.1 — Circles

1. 11 markNCERT Cl-10 Maths, Ex 10.1, Q1

   How many tangents can a circle have?

   Hint. Think of the tangent as touching at one point. How many points does a circle have?

   A tangent can be drawn at every single point of the circle, and a circle has infinitely many points on it — so there is no upper limit on how many tangents the circle as a whole can have.

   (This is different from asking how many tangents pass through one given point, which is answered in section 10.3 — 0, 1 or 2 depending on where that point sits.)

   ✦ Answer: infinitely many.

   Where students slip. Answering '2', which is the count of tangents from a single external point, not the count for the whole circle. The question asks about the circle, not about a point.

   Another way. Picture rotating the tangent line in Activity 1 all the way around the circle. It touches at a new point in every position, so no finite count works.

2. 22 marksNCERT Cl-10 Maths, Ex 10.1, Q2

   Fill in the blanks: (i) A tangent to a circle intersects it in ___ point(s). (ii) A line intersecting a circle in two points is called a ___. (iii) A circle can have ___ parallel tangents at the most. (iv) The common point of a tangent to a circle and the circle is called the ___.

   Hint. Each blank is one of the chapter's opening definitions. For (iii), picture two tangents on opposite sides of the circle, both parallel to the same line.

   (i) one. That is the definition of a tangent — exactly one common point with the circle.

   (ii) secant. A line meeting the circle in two points is a secant; extending a chord in both directions gives one.

   (iii) two. Given any direction, you can draw a tangent on each side of the circle parallel to that direction — one touching the 'near' side, one the 'far' side — and no third parallel tangent is possible, as Activity 2 shows.

   (iv) point of contact. This is the name given to the single shared point.

   ✦ Answer: (i) one (ii) secant (iii) two (iv) point of contact

   Where students slip. Answering (iii) with 'one', by picturing only a single tangent line and forgetting its mirror image on the opposite side of the circle.

   Another way. For (iii), imagine sliding a secant outward, parallel to itself, until it just touches the circle — this happens once on each side, giving two tangents, never more.

3. 31 markNCERT Cl-10 Maths, Ex 10.1, Q3

   A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q, so that OQ = 12 cm. Length PQ is: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm.

   Hint. P is the point of contact, so OP is a radius and OP ⊥ PQ. That makes OQ the hypotenuse of a right triangle.

   Step 1 — Identify the right angle. P is the point of contact of the tangent, so by Theorem 10.1, OP ⊥ PQ. This makes △OPQ right-angled at P, with OQ as the hypotenuse.

   Step 2 — Apply Pythagoras. OQ² = OP² + PQ² 12² = 5² + PQ² 144 = 25 + PQ² PQ² = 119

   Step 3 — Take the root. PQ = √119

   119 is not a perfect square (10² = 100, 11² = 121), so the answer stays as a surd — which is exactly option (D).

   ✦ Answer: (D) √119 cm

   Where students slip. Trying to match √119 to a familiar Pythagorean triple and forcing it to 12 or 13. Not every question is built from a triple — sometimes the surd itself is the intended answer, which the options confirm.

   Another way. A quick range check rules out two wrong options before computing anything: PQ must be less than OQ = 12 (since OQ is the hypotenuse), so 13 is impossible; and PQ must be more than OQ − OP = 7, so 8.5 is on the low side but worth checking properly rather than guessing.

4. 42 marksNCERT Cl-10 Maths, Ex 10.1, Q4

   Draw a circle and two lines parallel to a given line such that one is a tangent and the other a secant to the circle.

   Hint. Start from a line outside the circle and slide copies of it inward, parallel to the original, until one just touches and another cuts through.

   Step 1 — Draw a circle with centre O and any line ℓ that does not touch it.

   Step 2 — Draw a line parallel to ℓ, at a distance from the centre exactly equal to the radius. Because its distance from O equals the radius, it touches the circle at exactly one point — this is the tangent, parallel to ℓ.

   Step 3 — Draw a second line parallel to ℓ, at a distance from the centre that is less than the radius (for instance, passing close to O or through it). Since it is closer to the centre than the radius, it must cross into the circle and out again, meeting it at two points — this is the secant, also parallel to ℓ.

   ✦ Answer: one line at distance = radius from the centre gives the tangent; a second, closer line (distance < radius) gives the secant. Both are drawn parallel to the same given line ℓ.

   This is the essence of Activity 2 in the chapter — sliding parallel lines toward a circle, the last one to still miss it, the first to touch it, and every one after that becomes a secant.

   Where students slip. Drawing the 'secant' further from the centre than the tangent. Moving a parallel line *away* from the centre only ever produces non-intersecting lines or, at the boundary, the tangent — the secant must be drawn *closer* to the centre.

   Another way. Think in terms of the perpendicular distance d from the centre to the line, compared with the radius r: d > r gives a non-intersecting line, d = r gives the tangent, and d < r gives the secant. Choosing three parallel lines at three different distances shows all three cases at once.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 10 Mathematics textbook, Reprint 2026-27 (this chapter is unchanged by rationalisation — two exercises, 10.1 with 4 questions and 10.2 with 13, and the same two theorems in the summary). Questions are referenced from the NCERT textbook for identification.