Systems of Circles: Radical Axis, Common Tangents and Families

Name: Systems of Circles: Radical Axis, Common Tangents and Families
Author: StemPy Lab

Two round ponds overlap, and a fence is to run along the line through the two meeting points.

Where does that line lie?
Look at two circles together: how they touch, cut or miss, how many common tangents they have, and what their radical axis is.
Then meet orthogonal circles, the circles through the meeting points of two curves, and the locus questions that the entrance exams love.
It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and goes beyond the circle work of A-Level Pure Maths.

3 lessons, about 1 h 30 min3 simulations10 problems3 practice questionsPython in every lesson

Course outline (3 lessons)

Lesson 1about 30 min
Two circles and their common tangents
Two circles can be apart, touch, cross, touch from inside or lie one inside the other. Compare the distance $d$ between the centres with $r_1 + r_2$ and $|r_1 - r_2|$ and you know which case it is and how many common tangents there are: $4$ , $3$ , $2$ , $1$ or $0$ . The lengths of the common tangents are $\sqrt{d^2 - (r_1 \mp r_2)^2}$ .
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 30 min
Radical axis and orthogonal circles
The radical axis of two circles is the set of points whose tangent lengths to the two circles are equal. Its equation is $S_1 - S_2 = 0$ . Three circles have a radical centre where the three radical axes meet. Two circles are orthogonal when they cut at right angles, and then $2g_1g_2 + 2f_1f_2 = c_1 + c_2$ .
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 3about 30 min
Families of circles and locus problems
Many circles pass through two fixed points, and one number $\lambda$ picks each one. The circle $S + \lambda L = 0$ passes through the meeting points of the circle $S = 0$ and the line $L = 0$ , and $S_1 + \lambda S_2 = 0$ does the same for two circles. A locus is found by naming a general point $(h, k)$ , writing the condition and replacing $(h, k)$ by $(x, y)$ .
1 simulation1 practice question1 Python program4 problems1 question set1 note
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JEE Main format10 questions · 25 min

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