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NDA Circles

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This page covers NDA Circles with complete concept notes, 39 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Circles— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

A circle is the set of all points in a plane that are at equal distance from a fixed point called the center. In coordinate geometry, circles are represented by equations and can be analyzed using algebraic methods. The standard form of a circle's equation is (x-h)² + (y-k)² = r², where (h,k) is the center and r is the radius. This is the most

2.Formula Block

Memorise — at least one formula appears in every paper
for NDA exams. The general form is x² + y² + 2gx + 2fy + c = 0, where center is (-g,-f) and radius is √(g²+f²-c).

3.Key Rules

Core rules you must know cold
NDA papers frequently test

finding center and radius from given equation, equation of circle passing through three points, tangent and normal equations, and position of point relative to circle. Questions often involve finding intersection points with lines or other circles

SHORTCUT 1 - Quick Center-Radius

For x² + y² + 2gx + 2fy + c = 0, center is (-g,-f) and radius² = g²+f²-c. If g²+f²-c < 0, no real circle exists

SHORTCUT 2 - Tangent Length

From external point (x₁,y₁) to circle x² + y² + 2gx + 2fy + c = 0, tangent length = √(x₁² + y₁² + 2gx₁ + 2fy₁ + c) SHORTCUT 3 - Circle through three points: If you have three non-collinear points, substitute each in general form and solve the three equations simultaneously.

4.Worked Example

Solve this step-by-step before moving on
1
Step 1

Identify coefficients: 2g = -6, so g = -3; 2f = 4, so f = 2; c = -12

2
Step 2

Center = (-g,-f) = (3,-2)

3
Step 3

Radius = √(g²+f²-c) = √(9+4+12) = √25 = 5 Answer: Center (3,-2), radius 5 WORKED EXAMPLE 2: Find equation of circle with center (2,-3) and radius 4

1
Step 1

Use standard form (x-h)² + (y-k)² = r²

2
Step 2

Substitute h=2, k=-3, r=4

3
Step 3

(x-2)² + (y+3)² = 16

4
Step 4

Expanding: x²-4x+4 + y²+6y+9 = 16

5
Step 5

Simplifying: x² + y² - 4x + 6y - 3 = 0 Answer: x² + y² - 4x + 6y - 3 = 0

5.Exam Traps

Common mistakes students make — avoid these

#1: Students forget the negative sign when finding center from general form. Remember center is (-g,-f), not (g,f). This single error costs marks in 60% of circle problems.

Always double-check the signs when converting between forms. Another frequent error is calculating radius incorrectly from general form. Students often forget to check if g²+f²-c is positive. If it's negative or zero, the equation doesn't represent a real circle. For tangent problems, students mix up the condition for tangency.

A line is tangent to circle if perpendicular distance from center to line equals radius. Use the distance formula: |ax₀+by₀+c|/√(a²+b²) = r.

Key Points to Remember

  • Standard form: (x-h)² + (y-k)² = r² where (h,k) is center and r is radius
  • General form: x² + y² + 2gx + 2fy + c = 0 with center (-g,-f)
  • Radius formula from general form: r = √(g²+f²-c)
  • Tangent length from external point (x₁,y₁): √(x₁² + y₁² + 2gx₁ + 2fy₁ + c)
  • Equation of tangent at point (x₁,y₁): xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0
  • For tangency condition: distance from center to line = radius
  • Angle in semicircle is always 90 degrees
  • Tangent is perpendicular to radius at point of contact
  • Power of point formula: S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c
  • Circle equation needs g²+f²-c > 0 for real circle to exist

Exam-Specific Tips

  • Equation x² + y² = r² represents circle with center at origin and radius r
  • Parametric equations of circle: x = h + rcosθ, y = k + rsinθ
  • Length of tangent from origin to circle x² + y² + 2gx + 2fy + c = 0 is √c
  • Equation of circle on diameter with endpoints (x₁,y₁) and (x₂,y₂): (x-x₁)(x-x₂) + (y-y₁)(y-y₂) = 0
  • Condition for orthogonal circles: 2g₁g₂ + 2f₁f₂ = c₁ + c₂
  • Director circle of ellipse x²/a² + y²/b² = 1 is x² + y² = a² + b²
  • Radical axis of two circles is always a straight line
  • Family of circles through intersection of two circles: S₁ + kS₂ = 0
Practice MCQs

Circles — Practice Questions

39graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 39

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Practice 1easy

The circle x² + y² − 4x + 2y − 4 = 0 intersects the x-axis at two points. The distance between these two points is:

Practice 2easy

The equation of the tangent to the circle x² + y² = 25 at the point (3, 4) is:

Practice 3easy

The equation of the circle passing through the origin with centre at (3, 4) is:

Practice 4easy

The radius of the circle x² + y² − 4x + 6y − 3 = 0 is:

Practice 5easy

The centre and radius of the circle (x + 2)² + (y − 3)² = 49 are respectively:

Practice 6easy

If the circle x² + y² + 2gx + 2fy + c = 0 passes through the origin, then:

Practice 7easy

The general form of a circle is x² + y² − 8x + 6y + 9 = 0. What is the radius of this circle?

Practice 8easy

A circle passes through the points (1, 0) and (3, 0). If the centre lies on the line y = x, what is the radius of the circle?

Practice 9easy

The circle x² + y² = 16 is intersected by the line y = 2. At how many points do they intersect?

Practice 10easy

The equation of the circle with centre (3, −2) and radius 5 is:

Practice 11easy

The general form of a circle is x² + y² − 8x + 6y + 9 = 0. The radius of this circle is:

Practice 12easy

A circle passes through the points (1, 0) and (−1, 0), and its centre lies on the line y = x. The equation of the circle is:

Practice 13medium

A circle passes through the point (2, 3) and has its centre at (−1, 1). Find the equation of the circle in the form x² + y² + 2gx + 2fy + c = 0.

Practice 14medium

The circle x² + y² − 6x + 4y − 12 = 0 is touched by a line at the point (7, 2). Find the equation of the tangent line.

Practice 15medium

Two circles have equations x² + y² = 9 and (x − 4)² + (y − 3)² = 16. Find the length of the common chord (if it exists).

Practice 16medium

A circle with centre on the line 2x − y − 3 = 0 passes through the points (1, 2) and (3, 4). Find the radius of the circle.

Practice 17medium

The circle x² + y² − 4x − 6y + k = 0 is orthogonal to the circle x² + y² − 2x − 4y − 7 = 0. Find the value of k.

Practice 18medium

The circle x² + y² − 6x + 4y − 12 = 0 has center and radius respectively:

Practice 19medium

The equation of the tangent to the circle x² + y² = 25 at the point (3, 4) is:

Practice 20medium

Two circles have equations x² + y² − 2x − 4y − 4 = 0 and x² + y² − 6x − 8y + 16 = 0. The distance between their centers is:

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60-Second Revision — Circles

  • Remember: Center from general form is (-g,-f), not (g,f)
  • Formula: Radius = √(g²+f²-c) and check if positive
  • Trap: Always verify g²+f²-c > 0 for real circle
  • Quick: Tangent length = √(substitute point in circle equation)
  • Check: Distance from center to tangent line = radius
  • Pattern: Most NDA questions test center-radius finding and tangent problems
  • Time-saver: For three points, use determinant method for circle equation
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