Paper I — Mathematics

Circle

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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Core5 mapped NDA questions

Build the idea

Build exam-ready understanding by connecting the core idea, formulas or facts, worked example and previous NDA questions. Focus on why each rule works, then test it under timed conditions.

Concept

A circle is the locus of points equidistant (the radius) from a fixed point (the centre). Its equation comes in standard and general forms, from which the centre and radius are read off.

Many questions just need correct extraction of centre and radius.

Key Points & Formulas

Standard form: (x−h)² + (y−k)² = r², centre (h,k), radius r.
General form: x²+y²+2gx+2fy+c=0; centre (−g,−f), radius √(g²+f²−c).
A circle centred at the origin: x²+y² = r².
Tangency / position is judged by comparing distance-from-centre with r.

Worked Example

Q. Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.

Here 2g = −6 ⇒ g = −3; 2f = 4 ⇒ f = 2; c = −12.

Centre = (−g, −f) = (3, −2).

Radius = √(g²+f²−c) = √(9+4+12) = √25 = 5.

Common traps

Reading the prompt too quickly and missing a qualifier.
Memorising the result without checking when it applies.
Skipping units, signs, chronology or the final reasonableness check.

NDA shortcuts

State the governing rule before substituting values.
Eliminate impossible options before doing full calculation.

One-screen revision sheet

Recall before you practise

Core definition
Most-used rule
One representative example

Past-paper bank

Practise every mapped question

Filter by year

5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA I 2026

If the points lie on a circle, then what is/are the possible value(s) of $k$ ?

Question 2

NDA I 2025

A square is inscribed in a circle $x^2 + y^2 + 2x + 2y + 1 = 0$ and its sides are parallel to coordinate axes. Which one of the following is a vertex of the square?

Question 3

NDA I 2024

If (a, b) is the centre and c is the radius of the circle x² + y² + 2x + 6y + 1 = 0, then what is the value of a² + b² + c²?

Question 4

NDA II 2023

Consider the following in respect of the circle $4x^2 + 4y^2 - 4ax - 4ay + a^2 = 0$ :\n1. The circle touches both the axes.\n2. The diameter of the circle is $2a$ .\n3. The centre of the circle lies on the line $x + y = a$ .\nHow many of the statements given above are correct?

Question 5

NDA I 2023

What is the area of minor segment?

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