Overview of Circle in JEE Maths

The Circle is an important topic in JEE Maths coordinate geometry. Circles appear frequently in coordinate geometry, tangents, chords, and intersection problems. Mastering the circle helps students solve questions involving radius, center, tangents, chords, and circles passing through given points. At StudentBro.in, we provide a complete guide covering formulas, properties, examples, and practice problems to help JEE aspirants excel in the topic.

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Definition of a Circle

A circle is a set of all points in a plane equidistant from a fixed point called the center.

• Center: (h, k)

• Radius: r

Standard Equation of Circle:

(x - h)² + (y - k)² = r²

Example: Circle with center (2,3) and radius 5 → (x-2)² + (y-3)² = 25

General Equation of Circle

The general form of a circle:

x² + y² + 2gx + 2fy + c = 0

• Center: (-g, -f)

• Radius: √(g² + f² - c)

Example: x² + y² - 4x + 6y - 12 = 0 → Center = (2, -3), Radius = √(2² + (-3)² - (-12)) = √25 = 5

Equation of Circle with Given Center and Radius

If center (h, k) and radius r are given:

(x - h)² + (y - k)² = r²

Example: Center = (1, -2), Radius = 3 → (x-1)² + (y+2)² = 9

Circle Passing Through Three Points

If a circle passes through points (x₁, y₁), (x₂, y₂), (x₃, y₃), substitute in general equation to find g, f, and c.

• Solve system of three equations to get circle equation

• Frequently asked in JEE Advanced coordinate geometry

Example: Points (0,0), (2,0), (0,2) → x² + y² - 2x - 2y = 0

Tangent to a Circle

Equation of tangent at point P(x₁, y₁) on circle (x-h)² + (y-k)² = r²:

(x₁ - h)(x - h) + (y₁ - k)(y - k) = r²

• Slope form: y = mx + √(r²(1+m²)) ± intercept

Example: Circle x² + y² = 25, tangent at (3,4) → 3x + 4y = 25

Chord of a Circle

• Chord: A line segment connecting two points on a circle

• Length of chord: 2√(r² - d²), d = perpendicular distance from center to chord

Example: Circle x² + y² = 25, chord 4 units away from center → Length = 2√(25 - 16) = 6

Equation of Circle in Parametric Form

• Parametric coordinates: x = h + r cos θ, y = k + r sin θ

• Useful for intersection and tangent problems

Example: Circle x² + y² = 25 → x = 5 cos θ, y = 5 sin θ

Condition for Point on Circle

A point (x₁, y₁) lies on circle (x-h)² + (y-k)² = r² if:

(x₁ - h)² + (y₁ - k)² = r²

• Used in checking tangency, intersection, and locus problems

Intersection of Two Circles

• Two circles: (x-h₁)² + (y-k₁)² = r₁² and (x-h₂)² + (y-k₂)² = r₂²

• Solve simultaneously to find intersection points

• Can result in two points, one point (tangent), or no intersection

Example: x² + y² = 25 and (x-4)² + y² = 9 → Intersection points solved algebraically

Applications of Circle in JEE Maths

• Solve tangency and chord length problems

• Find equation with given center/radius or passing through points

• Solve intersections with lines or other circles

• Solve geometric problems in JEE Main & Advanced exams

• Used in mechanics, optics, and coordinate geometry applications

Tips and Tricks for JEE Circle

1. Memorize standard and general circle equations

2. Learn tangent and chord formulas

3. Practice finding circle passing through points

4. Draw diagrams to visualize intersections

5. Solve previous year JEE circle problems for speed and accuracy

StudentBro.in JEE Maths Circle Section

At StudentBro.in, students can access:

• Step-by-step explanations for equations, tangents, chords, and intersections

• Worked examples for circle passing through points and radius-center problems

• Practice questions with solutions and shortcuts

• Tips for quick problem-solving and exam strategy

• Revision notes for last-minute preparation

Practice Questions and Sample Problems

1. Find the equation of circle with center (2,3) and radius 5

2. Find tangent to circle x² + y² = 25 at point (3,4)

3. Find circle passing through points (0,0), (2,0), (0,2)

4. Length of chord 4 units away from center of circle x² + y² = 25

5. Intersection points of circles x² + y² = 25 and (x-4)² + y² = 9

Regular practice improves accuracy, speed, and confidence in solving circle problems in JEE exams.

Conclusion

Circle is a vital chapter in JEE Maths. Understanding equations, tangents, chords, parametric form, and intersections allows students to solve coordinate geometry problems efficiently. At StudentBro.in, we provide a complete guide from basics to advanced problem-solving, making circle problems simple, practical, and exam-oriented.

Mastering this topic ensures students can tackle tangents, intersections, and chord length problems confidently in both JEE Main and Advanced exams.