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Mathematics · Class 12 Science

Ch 11System of Circles — Class 12 Mathematics, concept-first.

When two circles intersect at a point , the angle between the circles at is defined as the angle between their two tangent lines at -- and since each tangent is perpendicular to the radius drawn to the point of contact, this is the same as the angle between the two radii and , where are the two centres.

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Key concepts

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Angle Between Two Intersecting Circles

The angle between two intersecting circles at a common point is defined as the angle between their tangents at -- equivalently (since each tangent is perpendicular to its own radius), the angle between the two radii and…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

2.1

Angle between Two Intersecting Circles

When two circles intersect at a point , the angle between the circles at is defined as the angle between their two tangent lines at -- and since each tangent is perpendicular to the radius drawn to th…

2.1.1

Orthogonal Circles

Two circles are orthogonal when they intersect at right angles, i.e. when the angle of the previous section equals .

2.1.2

Family of Circles through the Intersection of Two Circles

A single equation can describe an entire family of circles sharing the same pair of intersection points.

+Exercise 2(a)i8 questions
  1. Q1Find the angle between the circles $x^2+y^2-4x-6y-3=0$ and $x^2+y^2+2x+2y-2=0$.Free
  2. Q2Show that the circles $x^2+y^2-4x+2y+4=0$ and $x^2+y^2+6x-4y-20=0$ intersect orthogonally.Free
  3. Q3Find the value of $k$ so that the circles $x^2+y^2-6x-8y+12=0$ and $x^2+y^2+2kx-6y+3=0$ intersect orthogonally.Free
  4. Q4Find the equation and the length of the common chord of the circles $x^2+y^2-4x-2y+4=0$ and $x^2+y^2-2x-4y+4=0$.Preview
  5. Q5Find the equation of the circle passing through the origin and through the points of intersection of the circles $x^2+y^2-4x-6y-12=0$ and $x…Preview
  6. Q6Show that the circles $x^2+y^2-36=0$ and $x^2+y^2-6x-8y+24=0$ touch each other internally, and find the point of contact and the common tang…Preview
  7. Q7Find the equation of the circle described on the common chord of the circles $x^2+y^2-4x-2y+4=0$ and $x^2+y^2-2x-4y+4=0$ as diameter.Preview
  8. Q8Find the equation of the circle passing through the origin and through the points of intersection of the circle $x^2+y^2-4x-6y-12=0$ and the…Preview
2.2

Radical Axis of Two Circles

The power of a point with respect to a circle is the number obtained by substituting 's coordinates into the equation's left side; when lies outside the circle, equals the square of the length of the…

2.2.1

Radical Centre of Three Circles

Now consider three circles whose centres are not all on one line. Pairing them up gives three radical axes: that of (namely ), of (), and of ().

2.2.2

Circle Cutting Three Circles Orthogonally

The radical centre of three circles gives an immediate, direct construction of the unique circle that cuts all three of them orthogonally.

+Exercise 2(b)i5 questions
  1. Q1Find the radical axis of the circles $x^2+y^2-2x-4y-6=0$ and $x^2+y^2+4x-2y+1=0$.Free
  2. Q2Find the radical centre of the circles $x^2+y^2-4x-6y+5=0$, $x^2+y^2-2x-4y-1=0$ and $x^2+y^2+2x-2y-1=0$.Free
  3. Q3Find the equation of the circle which cuts the circles $x^2+y^2-4x-6y+5=0$, $x^2+y^2-2x-4y-1=0$ and $x^2+y^2+2x-2y-1=0$ orthogonally.Preview
  4. Q4Find the equation of the radical axis of the circles $x^2+y^2+4x-7=0$ and $2x^2+2y^2+3x+5y-9=0$.Preview
  5. Q5Show that the point $(3,1)$ lies on the radical axis of the circles $x^2+y^2-4x-6y+11=0$ and $x^2+y^2-2x-2y+1=0$, and show that the lengths…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 24 questions24 questions
  1. Q1If the angle between the circles $x^2 + y^2 - 12x - 6y + 41 = 0$ and $x^2 + y^2 + kx + 6y - 59 = 0$ is $45^\circ$, find $k$.Preview
  2. Q2Find the equation and length of the common chord of the two circles : $x^2 + y^2 + 3x + 5y + 4 = 0$ and $x^2 + y^2 + 5x + 3y + 4 = 0$.Preview
  3. Q3Show that the circles : $x^2 + y^2 - 6x - 9y + 13 = 0$, $x^2 + y^2 - 2x - 16y = 0$ touch each other. Find the point of contact and the equat…Preview
  4. Q4Find the equation of the radical axis of the circles $x^2 + y^2 + 4x + 6y - 7 = 0$, $4(x^2 + y^2) + 8x + 12y - 9 = 0$.Preview
  5. Q5Find the equation of the circle which passes through the point $(2, 0)$, $(0, 2)$ and orthogonal to the circle $2x^2+2y^2+5x-6y+4=0$.Preview
  6. Q6Find 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
  7. Q7Show that the equation of common tangents to the circle $x^2+y^2=2a^2$ and the parabola $y^2=8ax$ are $y=\pm(x+2a)$.Preview
  8. Q8Find the equation of the common chord of the circles : $x^2 + y^2 - 4x - 4y + 3 = 0$ and $x^2 + y^2 - 5x - 6y + 4 = 0$.Preview
  9. Q9Show that the angle between the circles $x^2 + y^2 = a^2$, $x^2 + y^2 = ax + ay$ is $\frac{3\pi}{4}$.Preview
  10. Q10Show that : $x^2 + y^2 - 6x - 9y + 13 = 0$, $x^2 + y^2 - 2x - 16y = 0$ touch each other. Find the point of contact and the equation of commo…Preview
  11. Q11Find the angle between the circles $x^2 + y^2 + 6x - 10y - 135 = 0$ and $x^2 + y^2 - 4x + 14y - 116 = 0$.Preview
  12. Q12Find the common tangent of the circles $x^2 + y^2 + 10x - 2y + 22 = 0$ and $x^2 + y^2 + 2x - 8y + 8 = 0$ at their point of contact.Preview
  13. Q13Find the equation of the circle which cuts orthogonally the circle $x^2 + y^2 - 4x + 2y - 7 = 0$ and having the centre at $(2, 3)$.Preview
  14. Q14Show 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
  15. Q15Find the equation of the circle which touches the circle $x^2 + y^2 - 2x - 4y - 20 = 0$ externally at $(5, 5)$ with radius 5.Preview
  16. Q16Find the equation of the circle passing through origin, having its centre on the line $x + y = 4$ and intersecting the circle $x^2 + y^2 - 4…Preview
  17. Q17Find K if the pair of circles $x^2 + y^2 + 4x + 8 = 0$, $x^2 + y^2 - 16y + k = 0$ are orthogonal.Preview
  18. Q18Find the radical centre of the 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
  19. Q19Show that $x^2 + y^2 - 6x - 9y + 13 = 0$, $x^2 + y^2 - 2x - 16y$ [... remainder of this circle's equation and a closing clause, most likely…Preview
  20. Q20Find the equation of the radical axis of the following circles : $x^2 + y^2 - 3x - 4y + 5 = 0$, $3(x^2 + y^2) - 7x + 8y - 11 = 0$.Preview
  21. Q21Find the equation of the circle which cuts orthogonally the circle $x^2 + y^2 - 4x + 2y - 7 = 0$ and having the center at $(2, 3)$.Preview
  22. Q22Find the equation of the common chord of the circles $x^2 + y^2 + 2x + 3y + 1 = 0$, $x^2 + y^2 + 4x + 3y + 2 = 0$.Preview
  23. Q23Find the equation of the circle which passes through the origin and intersects the circles $x^2 + y^2 - 4x - 6y - 3 = 0$, $x^2 + y^2 - 8y +…Preview
  24. Q24Find 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