Coordinate GeometryHigh Yield (2-3 Qs in JEE)

Circle & System of Circles Formulas

Standard circle, general equation, tangent conditions, chord of contact, director circle, and orthogonality.

9 Formulas•666+ Questions in Bank•

Concept Diagram

Circle Geometry, Tangents & Director Circle

Standard circle x² + y² = r² with tangent at point P(x₁,y₁), normal through origin, and concentric director circle x² + y² = 2r².

Core Formulas & Mathematical Identities

General Circle Equation & Radius

x^2 + y^2 + 2gx + 2fy + c = 0 \implies \text{Center: } (-g, -f), \quad r = \sqrt{g^2 + f^2 - c}

Standard parameters of general circle.

Examiner Tip: Always ensure the coefficients of x² and y² are 1 before identifying g, f, and c.

Tangent in Slope Form

y = mx \pm r\sqrt{1 + m^2}

Equation of tangent with slope m to circle x² + y² = r².

Examiner Tip: Condition of tangency: c² = r²(1 + m²).

Tangent at a Point (T = 0)

x x_1 + y y_1 + g(x + x_1) + f(y + y_1) + c = 0

Tangent equation at point P(x₁, y₁) lying on the circle.

Examiner Tip: Replace x² -> xx₁, y² -> yy₁, 2x -> (x + x₁), 2y -> (y + y₁).

Chord of Contact & Length of Tangent

\text{Chord of Contact: } T = 0, \quad \text{Length of Tangent: } L = \sqrt{S_1} = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}

Line joining contact points of tangents drawn from external point P(x₁, y₁).

Examiner Tip: Area of quadrilateral formed by tangents and radii = rL.

Condition of Orthogonal Intersection

2 g_1 g_2 + 2 f_1 f_2 = c_1 + c_2

Condition for two circles to cut each other at right angles (tangents are perpendicular).

Examiner Tip: Both circles must have coefficients of x² and y² equal to 1.

Director Circle of a Circle

x^2 + y^2 = 2 r^2

Locus of point of intersection of two perpendicular tangents to x² + y² = r².

Examiner Tip: Concentrical circle with radius √2 times the original circle radius.

Family of Circles & Radical Axis

S_1 + \lambda S_2 = 0 \quad (\lambda \ne -1), \quad S + \lambda L = 0, \quad \text{Radical Axis: } S_1 - S_2 = 0

Locus of points from which tangents drawn to two circles are of equal length. Perpendicular to line of centers.

Examiner Tip: Radical center is the point of concurrence of radical axes of three circles taken in pairs.

Lengths of Common Tangents

L_{\text{DCT}} = \sqrt{d^2 - (r_1 - r_2)^2}, \quad L_{\text{TCT}} = \sqrt{d^2 - (r_1 + r_2)^2} \quad (d = \text{center distance})

Direct and Transverse common tangents between two separated circles.

Examiner Tip: If d > r₁ + r₂: 4 common tangents; if d = r₁ + r₂: 3; if |r₁ - r₂| < d < r₁ + r₂: 2; if d = |r₁ - r₂|: 1; if d < |r₁ - r₂|: 0.

Chord with Given Midpoint & Length of Intercept

T = S_1, \quad \text{Chord Length } = 2\sqrt{r^2 - p^2}

Equation of chord of circle S = 0 bisected at point (x₁, y₁), where p is perpendicular distance from center to chord.

Examiner Tip: T = x x₁ + y y₁ + g(x+x₁) + f(y+y₁) + c, and S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Circle & System of Circles

•Radius requires g² + f² - c ≥ 0 (if negative, circle is imaginary).
•Forgetting that orthogonality condition 2g₁g₂ + 2f₁f₂ = c₁ + c₂ applies ONLY when coefficients of x² and y² are unity (1).

Fundamental Theory & Core Concepts

4 Topics

1Standard & General Equations of a Circle

›Geometric Definition: A circle is the locus of a point P(x, y) moving in a plane such that its distance from a fixed point (center) remains constant (radius r).
›Central Form: (x - h)² + (y - k)² = r² with center C(h, k) and radius r.
›General Equation: x² + y² + 2gx + 2fy + c = 0. Center is (-g, -f) and radius r = √(g² + f² - c). (1) Real circle if g² + f² - c > 0; (2) Point circle if g² + f² - c = 0; (3) Imaginary circle if g² + f² - c < 0.
›Diametric Form: If A(x₁, y₁) and B(x₂, y₂) are endpoints of a diameter, circle equation is (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0.

2Point Position & Power of a Point

›Position relative to S ≡ x² + y² + 2gx + 2fy + c = 0: Evaluate S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c.
›Point P(x₁, y₁) lies: OUTSIDE the circle if S₁ > 0; ON the circle if S₁ = 0; INSIDE the circle if S₁ < 0.
›Length of Tangent from external point P(x₁, y₁) is L = √S₁.
›Power of point P with respect to the circle is S₁ = L² = (CP)² - r².

3Tangents, Normals & Director Circle

›Condition of Tangency: A straight line y = mx + c touches the circle x² + y² = r² if and only if c² = r²(1 + m²).
›Point Form (T = 0): Tangent at point P(x₁, y₁) on the circle is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0.
›Normal: A line perpendicular to the tangent at the point of contact. CRUCIAL THEOREM: The normal to a circle at ANY point always passes through its center (-g, -f).
›Director Circle: The locus of the point of intersection of two mutually perpendicular tangents to a circle. It is a concentric circle with radius r_dir = √2 · r: x² + y² = 2r².

4Chords & Relative Position of Two Circles

›Chord of Contact: The line joining the contact points of two tangents drawn from an external point P(x₁, y₁) is given by T = 0.
›Chord Bisected at a Given Point: If midpoint of chord is M(x₁, y₁), its equation is T = S₁.
›Relative Positions of Circles with centers C₁, C₂ and radii r₁, r₂ (d = C₁C₂): (1) Disjoint outside: d > r₁ + r₂ (4 common tangents); (2) Touching externally: d = r₁ + r₂ (3 common tangents); (3) Intersecting: |r₁ - r₂| < d < r₁ + r₂ (2 common tangents); (4) Touching internally: d = |r₁ - r₂| (1 common tangent); (5) One inside other: d < |r₁ - r₂| (0 common tangents).
›Orthogonal Circles: Two circles intersect orthogonally if tangents at their point of intersection are perpendicular: 2g₁g₂ + 2f₁f₂ = c₁ + c₂.

Previous Chapter

Straight Lines & Pair of Lines

7 Formulas•Coordinate Geometry

Next Chapter

Parabola & Conics

Coordinate Geometry•8 Formulas