Summary of "Circle | One Shot | #BounceBack Series | Unacademy Atoms | JEE Maths | Nishant Vora"

Overview

Instructor: Nishant Vora (Unacademy Atoms, BounceBack series).
Class type: One‑shot revision of the Circle chapter for JEE Main / Advanced.
Focus: definitions, standard equations, common question types, fast contest methods, and many PYQ‑style worked examples (Mains & Advanced 2019–2021 cited).
Emphasis: clear conceptual understanding + speed. Tips on resources, revision strategy (repeat a limited set of high‑value material), and exam mindset.

Basic definition - A circle is the locus of a point at a fixed distance (radius r) from a fixed point (centre (a, b)).
Standard forms of circle equations - Centre–radius form: (x − a)^2 + (y − b)^2 = r^2

Centre at origin (0,0): x^2 + y^2 = r^2
General form: x^2 + y^2 + 2gx + 2fy + c = 0
- centre = (−g, −f)
- radius = sqrt(g^2 + f^2 − c)

How to get centre & radius from general form - Compare coefficients or complete the square. - Shortcut: centre = (−g, −f), r = sqrt(g^2 + f^2 − c).
Equation forms you will use (and when) - Diameter endpoints A(x1,y1), B(x2,y2): (x − x1)(x − x2) + (y − y1)(y − y2) = 0 (faster than computing midpoint + radius)

Three‑point form: Substitute three points into x^2 + y^2 + 2gx + 2fy + c = 0 → 3 linear equations → solve for g, f, c.
Parametric coordinates (eccentric angle θ): For centre (a, b), radius r: P = (a + r cos θ, b + r sin θ). For centre = origin: (r cos θ, r sin θ).

Position of a point relative to a circle (quick test) - For point (x1, y1), compute S1 = x1^2 + y1^2 + 2g x1 + 2f y1 + c.
- S1 > 0 → point outside
- S1 = 0 → point on the circle
- S1 < 0 → point inside

Quick test (used repeatedly): evaluate S1; its sign tells you inside / on / outside.

Line–circle intersection (method & conditions) - Substitute the line into the circle → quadratic. Use discriminant D:
- D > 0 → secant (2 intersections)
- D = 0 → tangent (1 intersection)
- D < 0 → no real intersection
- Geometric equivalent: perpendicular distance d from centre to the line:
- d < r → 2 intersections
- d = r → tangent
- d > r → no intersection
Tangent equations (three common forms) - Point form (if tangent at (x1, y1) on circle): use the replacement rules (T = 0). - Parametric form (tangent at θ): for centre origin, x x1 + y y1 = r^2 (shift for centre ≠ origin). - Slope form (slope m): shifted form y − b = m(x − a) ± r sqrt(1 + m^2) (two parallel tangents with same m in general).
Tangents from an external point, pair of tangents, chord of contact - If P(x1, y1) is exterior (S1 > 0), two tangents can be drawn. - Pair of tangents combined equation: T · S = S1 (S = circle polynomial; T = linear form from substitutions)

Chord of contact (join of contact points): T = 0.
If (h, k) is midpoint of a chord: chord equation = T = S1 (midpoint substitution utility).

Length of tangent and power of a point - Length of tangent from P = sqrt(S1). - Power of point = S1 (can be positive, negative, zero).
Director circle - Locus of points from which the pair of tangents are perpendicular. - For (x − a)^2 + (y − b)^2 = r^2, director circle: (x − a)^2 + (y − b)^2 = 2 r^2
- centre same as original; radius = √2 · r.
Useful triangle/geometry facts - Radius ⟂ tangent at point of contact. - Perpendicular from centre bisects the chord. - Angle in a semicircle is 90° (right angle implies diameter). - Use similarity (AA) of right triangles formed by radii and tangents.

Write circle from centre (a, b) & radius r: 1) Use (x − a)^2 + (y − b)^2 = r^2; expand if required.
Find centre & radius from general equation: 1) Identify 2g, 2f coefficients → centre (−g, −f). 2) r = sqrt(g^2 + f^2 − c).
Tangent at point (x1, y1) on circle: 1) Verify S1 = 0. 2) Use point form (T = 0), parametric substitution or slope form as appropriate.
Tangents from external point P(x1, y1): 1) Slope method: y = m x + c, impose passing through P and tangency condition (discriminant = 0 or perpendicular distance = r); solve for m. 2) Or use combined form T·S = S1 to get pair of tangents directly.
Pair of tangents / chord of contact:
- Pair: T·S = S1
- Chord of contact: T = 0
- Chord with midpoint (h, k): T = S1
Test point vs circle:
- Compute S1 and use sign test.
Line–circle tangency check:
- Use perpendicular distance or discriminant.
Use parametric substitution (x = a + r cos θ, y = b + r sin θ) to reduce two variables to one in optimization problems.

Converting general equation to centre & radius.
Circle with given diameter endpoints.
Circle through three given points (solve for g, f, c).
Tangent through a given point; tangent parallel to a given line.
Finding diameter or chord length when chord is diameter of another circle.
PYQs: midpoint locus of chord of contact; tangent/normal area problems; maximizing sums of distances using parametric form; tangency derived from discriminant.
Director circle problems.
Power of a point & length of tangent calculations.

Use diameter form when diameter endpoints are given — it’s faster.
For a known point on the circle, use T = 0 (replacement trick) for quick tangents.
For exterior points and tangents, prefer slope/discriminant or T·S = S1 rather than point‑on‑circle shortcuts.
When squaring equations, be mindful of losing sign information — often ± corresponds to symmetric tangents, but check context.
Prefer parametric form to reduce variables in optimization problems.
If stuck, draw a neat diagram — many algebraic problems are geometric in disguise.

Caution: Do not casually mix point‑on‑circle shortcuts with exterior‑point tangent constructions; check S1 first to decide which method applies.

Repeated viewing allowed — watch one‑shot twice if needed to achieve clarity.
Use Unacademy Atoms for recorded lectures and PDFs; Unacademy Plus for structured courses, doubt classes, and test series.
Instructor advised consistent practice on PYQs and repeating a small set of core concepts rather than superficially covering many topics.
Instructor mentioned platform promo codes (check the platform for current codes).

Main speaker: Nishant Vora (Unacademy Atoms / Unacademy — BounceBack series).
Sources: JEE Main & JEE Advanced previous‑year questions (PYQs), Unacademy Atoms recordings, Unacademy Plus.

One‑page cheat sheet of frequently used formulas and 6–8 recipes (conversion, tangent forms, pair‑of‑tangents, chord formulas, point test, discriminant test).
Reworked step‑by‑step examples from lecture (e.g., locus/midpoint of chord of contact problems).