Is System of circles part of the EAPCET Mathematics syllabus?

Yes. System of circles is taught in Intermediate second year · Paper IIB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.

How many Mathematics questions are there in EAPCET?

80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.

What mistakes do students make in System of circles?

Applying 2g₁g₂ + 2f₁f₂ = c₁ + c₂ before both equations have x² and y² coefficients equal to 1. Subtracting circle equations for the radical axis without first normalising both; S − S′ = 0 needs the same leading coefficients. Mixing up the tangent counts; touching externally gives 3 common tangents and touching internally gives only 1. Writing 2g₁g₂ + 2f₁f₂ = c₁ − c₂ or dropping the factor 2; the orthogonality condition is 2g₁g₂ + 2f₁f₂ = c₁ + c₂.

EAPCET Mathematics · Intermediate second year · Paper IIB

System of circles

System of Circles studies two or more circles together: the angle between them, orthogonal circles, radical axis and centre, and common tangents.

Subject: Mathematics
Taught in: Intermediate second year · Paper IIB
Mathematics in the paper: 80 of 160 questions

Key formulas and results

Orthogonal circles: $2g_{1}g_{2} + 2f_{1}f_{2} = c_{1} + c_{2}$
Radical axis: $S - S' = 0$
Angle between circles: $\cos\theta = \dfrac{r_{1}^{2} + r_{2}^{2} - d^{2}}{2r_{1}r_{2}}$
Touch externally when $d = r_{1} + r_{2}$ , internally when $d = |r_{1} - r_{2}|$
Common tangents: 4 if separate, 3 if touching externally, 2 if intersecting, 1 if touching internally, 0 if one lies inside the other
Length of the direct common tangent of two circles: $\sqrt{d^{2} - (r_{1} - r_{2})^{2}}$ , where $d$ is the distance between centres
Length of the transverse common tangent: $\sqrt{d^{2} - (r_{1} + r_{2})^{2}}$
Orthogonal circles in terms of radii: $r_{1}^{2} + r_{2}^{2} = d^{2}$
Radical centre of three circles: the common solution of $S_{1} - S_{2} = 0$ and $S_{2} - S_{3} = 0$

What this chapter covers

Angle between two circles
Orthogonal circles
Radical axis and radical centre
Coaxal systems
Common tangents

How it is usually asked

Test orthogonality
Find a radical axis or radical centre
Count common tangents

Understanding System of circles

When two circles are studied together, the key numbers are their radii r₁, r₂ and the distance d between their centres. Comparing d with r₁ + r₂ and |r₁ − r₂| tells you everything about their position: separate, touching externally, intersecting, touching internally or one inside the other. The number of common tangents follows at once: 4, 3, 2, 1 or 0 in that order.

Intersecting circles meet at an angle θ, the angle between their tangents at a common point, given by cos θ = (r₁² + r₂² − d²)/(2r₁r₂). When θ = 90° the circles are orthogonal, which happens when d² = r₁² + r₂², or in coefficient form 2g₁g₂ + 2f₁f₂ = c₁ + c₂. This condition is the quickest test and is used to find unknown constants.

The radical axis of two circles S = 0 and S′ = 0 is the line S − S′ = 0, the set of points from which the tangents to both circles have equal length. For intersecting circles it is the common chord. The radical axes of three circles taken in pairs meet at the radical centre. EAPCET questions ask for these lines and points, orthogonality conditions and common tangent counts.

Worked example

EAPCET-style question

The number of common tangents to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 6x - 8y + 16 = 0$ is

(1)1
(2)2
(3)3
(4)4

Solution

First circle: centre $(0, 0)$ , radius 2.
Second circle: centre $(3, 4)$ , radius $\sqrt{9 + 16 - 16} = 3$ .
Distance between centres $d = \sqrt{9 + 16} = 5 = 2 + 3 = r_{1} + r_{2}$ .
The circles touch externally, so they have 3 common tangents.

Answer: 3

Shortcuts and results to remember

Compare $r_{1} + r_{2}$ with $d$ : if $r_{1} + r_{2} < d$ the circles are separate, if equal they touch externally, if greater they cut (for $d > |r_{1} - r_{2}|$ ).

Common mistakes to avoid

Applying 2g₁g₂ + 2f₁f₂ = c₁ + c₂ before both equations have x² and y² coefficients equal to 1.
Subtracting circle equations for the radical axis without first normalising both; S − S′ = 0 needs the same leading coefficients.
Mixing up the tangent counts; touching externally gives 3 common tangents and touching internally gives only 1.
Writing 2g₁g₂ + 2f₁f₂ = c₁ − c₂ or dropping the factor 2; the orthogonality condition is 2g₁g₂ + 2f₁f₂ = c₁ + c₂.

How to prepare it

Compare the distance between centres with r₁ + r₂ and |r₁ − r₂| first — it settles most positional questions.

Once the formulas feel familiar, test them the way the exam does: practise System of circles questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise System of circles chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is System of circles part of the EAPCET Mathematics syllabus?: Yes. System of circles is taught in Intermediate second year · Paper IIB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare System of circles for EAPCET?: Compare the distance between centres with r₁ + r₂ and |r₁ − r₂| first — it settles most positional questions.
How many Mathematics questions are there in EAPCET?: 80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in System of circles?: Applying 2g₁g₂ + 2f₁f₂ = c₁ + c₂ before both equations have x² and y² coefficients equal to 1. Subtracting circle equations for the radical axis without first normalising both; S − S′ = 0 needs the same leading coefficients. Mixing up the tangent counts; touching externally gives 3 common tangents and touching internally gives only 1. Writing 2g₁g₂ + 2f₁f₂ = c₁ − c₂ or dropping the factor 2; the orthogonality condition is 2g₁g₂ + 2f₁f₂ = c₁ + c₂.