Circle – Complete JEE Notes (Concepts + Formulae + Tricks)
A circle looks innocent until JEE throws slopes, tangents, radicals, and distances at you. This guide softens every sharp edge by explaining things visually and cleanly. All formulas are MathJax-formatted for perfect rendering.
1. Basic Definition
A circle is the locus of points that are at a fixed distance (radius) from a fixed point (center).
Standard Form
Center $(a, b)$ and radius $r$:
$$ (x - a)^2 + (y - b)^2 = r^2 $$
General Form
$$ x^2 + y^2 + 2gx + 2fy + c = 0 $$ Center $C(-g, -f)$ Radius: $$ r = \sqrt{g^2 + f^2 - c} $$ Condition for real circle: $$ g^2 + f^2 - c > 0 $$
2. Pain Points – Solved Clearly
⭐ Pain Point 1: “How do I check if a line intersects a circle?”
Compute perpendicular distance of center from line. If $D < r$ → secant If $D = r$ → tangent If $D > r$ → no intersection
⭐ Pain Point 2: “How do I find equation of tangent quickly?”
Use this golden template:
**Tangent at $(x_1, y_1)$:** $$ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 $$
Or, if center is $(a, b)$ and $P(x_1, y_1)$:$$ (x - a)(x_1 - a) + (y - b)(y_1 - b) = r^2 $$
3. Common Results Students Usually Forget
Distance of a Point from Circle
For point $P(x_1, y_1)$: $$ d = |CP - r| $$ Inside if $CP < r$, On if $CP = r$, Outside if $CP > r$.
Length of Tangent from Point
For $P(x_1, y_1)$ to circle: $$ PT = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c} $$
Radical Axis
Difference of circle equations gives: $$ (S_1 - S_2 = 0) $$ A straight line. Every point on it has equal power w.r.t both circles.
Power of a Point
$$ \text{Power} = PT^2 $$
4. Circle Terminology (All in One Place)
Chord: Line segment between two points on circle Diameter: Largest chord, $2r$ Arc: Portion of circumference Sagitta: Height of an arc Sector: Area: $A = \frac{\theta}{360^\circ} \pi r^2$ Segment: Area between chord and arc Concentric Circles: Same center, different radii Orthogonal Circles: Intersect at right angles Condition: $$ 2(g_1 g_2 + f_1 f_2) = c_1 + c_2 $$
5. Parametric Form (Super Useful for JEE)
For $(x - a)^2 + (y - b)^2 = r^2$:
$$ x = a + r\cos\theta $$ $$ y = b + r\sin\theta $$
Used in differentiability, tangents, locus problems.6. Director Circle
Locus of midpoints of all chords perpendicular to each other: $$ x^2 + y^2 = 2r^2 $$ (center at origin)
7. Equation of Chord With Midpoint Known
If circle is $x^2 + y^2 = r^2$ and midpoint is $(x_1, y_1)$: $$ xx_1 + yy_1 = r^2 $$
8. Tangent and Normal Through Parametric Point
For $P(r\cos\theta, r\sin\theta)$:
Tangent: $$ x\cos\theta + y\sin\theta = r $$ Normal: $$ \frac{x}{\cos\theta} = \frac{y}{\sin\theta} $$
9. Family of Circles (Locus Problems)
Circles Through Two Points
General: $$ S_1 + \lambda S_2 = 0 $$
Circle Touching a Line
Condition: distance from center = radius.
10. All Important Formulae at a Glance
Radius in general form: $$ r = \sqrt{g^2 + f^2 - c} $$ Distance from center to line $ax+by+c=0$: $$ D = \frac{|a(-g) + b(-f) + c|}{\sqrt{a^2 + b^2}} $$ Condition for tangency: $$ D = r $$ Power of point: $$ S = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c $$ Chord length formula (from perpendicular distance $d$): $$ \ell = 2\sqrt{r^2 - d^2} $$
11. JEE-Trap Questions Students Often Miss
Trap 1: A line may look tangent but actually misses the circle. Always check distance = radius.
Trap 2: “Minimum distance between two circles” Use center distance:
$$ d_{\min} = |C_1C_2 - (r_1 + r_2)| $$
Trap 3: Radical axis questions often hide pair of intersecting lines.
12. Tiny Illustration Problems
Q1. Length of chord cut by line $x = 3$ in circle $x^2 + y^2 = 25$
Distance from center $(0,0)$ to line is $3$. Chord length: $$ 2\sqrt{25 - 9} = 8 $$
Q2. Find the tangent from point $(4,3)$ to circle $x^2 + y^2 = 9$
Power = $4^2 + 3^2 - 9 = 16$ Length of tangent = $4$ Equation: $$ 4x + 3y = 16 $$
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