Mathematics · Class 12 Science
Ch 12System of Circles — Class 12 Mathematics, concept-first.
So far you've studied a single circle at a time — its equation, its tangent, its normal, its pole and polar. This chapter is about two (or three) circles together, and the relationships that appear when they share the plane.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
System of Circles
System of Circles studies how two or three circles relate to each other: the angle at which two circles cross (and the special right-angle case of orthogonal circles), the radical axis of equal power that becomes their c…
Most relevant Q&A
- Find k if the pairs of circles $x^2+y^2+4x+8=0$, $x^2+y^2-16y+k=0$ are orthogonal.Preview
- Show that the circles $x^2+y^2-8x-2y+8=0$ and $x^2+y^2-2x+6y+6=0$ touch each other and find the point of contact.Preview
- Find the transverse common tangents of the circles $x^2+y^2-4x-10y+28=0$ and $x^2+y^2+4x-6y+4=0$.Preview
- Find the radical centre of the following circles. $x^2 + y^2 - 4x - 6y + 5 = 0$, $x^2 + y^2 - 2x - 4y - 1 = 0$, $x^2 + y^2 - 6x - 2y = 0$.Preview
- Show that the circles $x^2 + y^2 - 6x - 2y + 1 = 0$, $x^2 + y^2 + 2x - 8y + 13 = 0$ touch each other. Find the point of contact and the equa…Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction: What is a System of Circles?
So far you've studied a single circle at a time — its equation, its tangent, its normal, its pole and polar.
Angle Between Two Intersecting Circles
Setting up the idea. Two circles intersect when the distance between their centres is neither too large (circles too far apart) nor too small (one swallowed inside the other) — precisely when , where…
Orthogonal Circles
Definition. Two intersecting circles are called orthogonal if the angle between them (as defined in the previous section) is exactly — the two tangent lines at the intersection point are perpendicular…
The Radical Axis of Two Circles
Power of a point, recalled. For a circle and any point , the quantity is called the power of with respect to the circle.
Common Chord and the Radical Centre of Three Circles
The common chord is the radical axis. Suppose two circles and actually intersect at two distinct points (rather than just having some abstract locus of equal power).
Intersection of a Line with a Circle: the Family S + λL = 0
The idea. Suppose a line cuts a circle at two points and . There are infinitely many other circles that also pass through the same two points — picture rotating a circle of varying size through the fi…
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Find k if the pairs of circles $x^2+y^2+4x+8=0$, $x^2+y^2-16y+k=0$ are orthogonal.Preview
- Q2Show that the circles $x^2+y^2-8x-2y+8=0$ and $x^2+y^2-2x+6y+6=0$ touch each other and find the point of contact.Preview
- Q3Find the transverse common tangents of the circles $x^2+y^2-4x-10y+28=0$ and $x^2+y^2+4x-6y+4=0$.Preview
- Q4Find the radical centre of the following circles. $x^2 + y^2 - 4x - 6y + 5 = 0$, $x^2 + y^2 - 2x - 4y - 1 = 0$, $x^2 + y^2 - 6x - 2y = 0$.Preview
- Q5Show that the circles $x^2 + y^2 - 6x - 2y + 1 = 0$, $x^2 + y^2 + 2x - 8y + 13 = 0$ touch each other. Find the point of contact and the equa…Preview