NCERT Solutions for Class 9th Maths Chapter 5 Think and Reflect — How Many Circles?
Book page 95 Updated on2026-09-08
Q1.
How many circles pass through two points on a plane?
Answer
Infinitely many.
Why it happens: a circle through A and B must have its centre O at a point with OA = OB, i.e. on the perpendicular bisector of AB. Conversely, every point O of that perpendicular bisector gives a circle: take radius OA, and since OB = OA the circle passes through B too. The perpendicular bisector contains infinitely many points, so there are infinitely many such circles — one for each point of the line.
Every point of the perpendicular bisector of AB is the centre of a circle through A and B.
Q2.
Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?
Answer
No — not all radii are possible. Only radii of at least half of AB occur.
Let AB = c and let the centre O be at distance h from the midpoint M of AB.
Then OA² = OM² + MA², i.e. r² = h² + (c/2)²
So r = √(h² + c²/4) ≥ c/2 for every h ≥ 0.
Smallest radius = c/2 = ½ AB, obtained when h = 0, i.e. when the centre is the midpoint of AB. For that circle AB is a diameter.
Largest radius: there is none. As h increases, r increases without bound, so the radii can be made as large as we like but never reach a maximum.
Why it happens: AB is a chord of every such circle, and no chord can be longer than the diameter. So 2r ≥ AB always, which is exactly r ≥ ½AB. Radii smaller than ½AB are ruled out; every radius from ½AB upwards actually occurs.
Q3.
As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease?
Answer
The radii increase.
r = √(h² + (AB/2)²) where h = distance of the centre from the midpoint of AB
h ↑ ⇒ h² ↑ ⇒ h² + (AB/2)² ↑ ⇒ r ↑
Why it happens: A stays on the circle, so the radius is the distance OA. Moving O further away from AB along the perpendicular bisector simply moves it further from A as well — the right triangle OMA gets a longer vertical leg while its horizontal leg MA stays fixed at ½AB, so the hypotenuse OA grows.
Q4.
As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved?
Answer
Less curved — the arc through A and B gets flatter and flatter.
Why it happens: curvature is decided by the radius: a small circle bends sharply, a large circle bends gently. From Q3 the radius grows as you move out along the perpendicular bisector, so near A and B the arc becomes flatter. In the limit, as the centre races off to infinity, the arc through A and B straightens into the line AB itself.
Did you know? This is why the horizon looks straight. You are standing on a circle of radius about 6400 km, so over a few metres the curvature is far too gentle to notice.
Q5.
You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square?
Answer
(a) A and B anywhere on the boundary: infinitely many squares.
Draw the segment AB. For any square you like, you can slide and turn it until its boundary passes through both A and B — for instance take any line through A and any line through B, and there are squares of many different sizes and tilts meeting both. There is no bound on the size either, so there are infinitely many.
(b) A and B as corners (vertices) of the square: exactly 3 squares.
Case 1 — AB is a side: the square can be built on either side of AB → 2 squares
Case 2 — AB is a diagonal: the other two vertices are fixed, one on each side of AB → 1 square
Total = 2 + 1 = 3 squares
With A and B as vertices there are only three squares — two with AB as a side, one with AB as a diagonal.
Why the sharp contrast: "on the boundary" leaves the size and the tilt free, so there is a whole family of answers. "As corners" fixes one full side or one full diagonal, and a square is completely determined once you know a side (plus which way to build it) or a diagonal.