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Mathematics · Class 11 Science

Ch 6Circle — Class 11 Mathematics, concept-first.

A circle is the set of all points in a plane that lie at one fixed distance from one fixed point. That fixed point is called the centre, and the fixed distance is called the radius of the circle.

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General Equation of a Circle

Expanding the centre-radius form and matching coefficients with shows that every circle can be written in this "general" second-degree form, with centre and radius .

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Introduction

A circle is the set of all points in a plane that lie at one fixed distance from one fixed point. That fixed point is called the centre, and the fixed distance is called the radius of the circle.

6.1

Different Forms of Equation of a Circle

This section develops three equivalent ways to write the equation of a circle, depending on what information a problem gives you:

+Exercise 6.1i18 questions
  1. Q1Find the equation of the circle with centre at origin and radius $4$.Free
  2. Q2Find the equation of the circle with centre at $(-3,-2)$ and radius $6$.Free
  3. Q3Find the equation of the circle with centre at $(2,-3)$ and radius $5$.Free
  4. Q4Find the equation of the circle with centre at $(-3,-3)$ passing through the point $(-3,-6)$.Preview
  5. Q5Find the centre and radius of the circle $x^2+y^2=25$.Preview
  6. Q6Find the centre and radius of the circle $(x-5)^2+(y-3)^2=20$.Preview
  7. Q7Find the centre and radius of the circle $\left(x-\dfrac12\right)^2+\left(y+\dfrac13\right)^2=\dfrac{1}{36}$.Preview
  8. Q8Find the equation of the circle with centre at $(a,b)$ touching the Y-axis.Preview
  9. Q9Find the equation of the circle with centre at $(-2,3)$ touching the X-axis.Preview
  10. Q10Find the equation of the circle with centre on the X-axis and passing through the origin, having radius $4$.Preview
  11. Q11Find the equation of the circle with centre at $(3,1)$ and touching the line $8x-15y+25=0$.Preview
  12. Q12Find the equation of the circle if the equations of two diameters are $2x+y=6$ and $3x+2y=4$, when the radius of the circle is $9$.Preview
  13. Q13If $y=2x$ is a chord of the circle $x^2+y^2-10x=0$, find the equation of the circle with this chord as diameter.Preview
  14. Q14Find the equation of a circle with radius $4$ units and touching both the co-ordinate axes, having centre in the third quadrant.Preview
  15. Q15Find the equation of the circle (a) passing through the origin and having intercepts $4$ and $-5$ on the co-ordinate axes.Preview
  16. Q16Find the equation of a circle passing through the points $(1,-4)$, $(5,2)$ and having its centre on the line $x-2y+9=0$.Preview
  17. Q17Construct a circle in the fourth quadrant having radius $3$ and touching the Y-axis. How many such circles can be drawn?Preview
  18. Q18Construct a circle whose equation is $x^2+y^2-4x+6y-12=0$. Find the area of the circle.Preview
6.1.1

Standard Form (Centre at Origin)

Let the centre of the circle be the origin and let the radius be . Let be any point lying on the circle. Because is on the circle, its distance from the centre must equal the radius:

6.1.2

Centre-Radius Form

Now let the centre be any point , not necessarily the origin, and let the radius still be . Let be any point on the circle, so

6.1.3

Diameter Form

Setting up the picture. Let and be the two endpoints of a diameter of a circle, so the centre is the midpoint of . Let be any other point on the circle.

6.2

General Equation of a Circle

From centre-radius form to a standard second-degree pattern. Start from the centre-radius form with centre and radius :

+Exercise 6.2i6 questions
  1. Q19Find the centre and radius of the circle $x^2+y^2-2x+4y-4=0$.Free
  2. Q20Find the centre and radius of the circle $x^2+y^2-6x-8y-24=0$.Free
  3. Q21Find the centre and radius of the circle $4x^2+4y^2-24x-8y-24=0$.Preview
  4. Q22Show that the equation $3x^2+3y^2+12x+18y-11=0$ represents a circle.Preview
  5. Q23Find the equation of the circle passing through the points $(5,7)$, $(6,6)$ and $(2,-2)$.Preview
  6. Q24Show that the points $(3,-2)$, $(1,0)$, $(-1,-2)$ and $(1,-4)$ are concyclic.Preview
6.3

Parametric Form of a Circle

Setting up the angle parameter. Let be any point on a circle centred at the origin with radius . Let be the angle that makes with the positive direction of the X-axis.

+Exercise 6.3i7 questions
  1. Q28Write the parametric equations of the circle $x^2+y^2=9$.Free
  2. Q29Write the parametric equations of the circle $x^2+y^2+2x-4y-4=0$.Free
  3. Q30Write the parametric equations of the circle $(x-3)^2+(y+4)^2=25$.Free
  4. Q31Find the parametric representation of the circle $3x^2+3y^2-4x+6y-4=0$.Preview
  5. Q32Find the equation of a tangent to the circle $x^2+y^2-3x+2y=0$ at the origin.Preview
  6. Q33Show that the line $7x-3y-1=0$ touches the circle $x^2+y^2+5x-7y+4=0$ at the point $(1,2)$.Preview
  7. Q34Find the equation of the tangent to the circle $x^2+y^2-4x+3y+2=0$ at the point $(4,-2)$.Preview
6.3.1

Tangent to a Circle

Definition. When a line meets a circle at two coincident points (rather than two distinct points, or none), it is called a tangent to the circle, and the single point where they meet is the point of c…

6.3.2

Condition of Tangency

Goal. Find the condition under which a general line (a line of unknown slope and intercept , not tied to a specific point) is a tangent to the circle , and find the point of contact in terms of and .

6.3.3

Tangents from an External Point to a Circle

Setting up. Let be a point in the plane, lying outside the circle (so it is not on the circle itself). Suppose a tangent line from has slope ; by the point-slope form, its equation is

6.3.4

Director Circle

Definition. The director circle of a given circle is the locus (path traced) of the point of intersection of a pair of mutually perpendicular tangents drawn to that circle.

6.4

Relative Positions of Two Circles (Extra Information)

Given two circles, with centres and radii , comparing the distance between their centres against and tells you exactly how the two circles sit relative to each other, and how many lines can be drawn t…

More questions

44 Q
+Show 3 questions3 questions
  1. Activity 6.2Check whether the equation $x^2+y^2-6x-4y+9=0$ represents a circle. If so, find its centre and radius.Free
  2. Activity 6.2Check whether the equation $x^2+y^2-8x+6y+29=0$ represents a circle. If so, find its centre and radius.Preview
  3. Activity 6.2Check whether the equation $x^2+y^2+7x-5y+15=0$ represents a circle. If so, find its centre and radius.Preview
+Show 10 questions10 questions
  1. Q35Equation of a circle which passes through $(3,6)$ and touches the axes is (A) $x^2+y^2+6x+6y+3=0$ (B) $x^2+y^2-6x-6y-9=0$ (C) $x^2+y^2-6x-6y…Free
  2. Q36If the lines $2x-3y=5$ and $3x-4y=7$ are the diameters of a circle of area $154$ sq. units, then find the equation of the circle. (A) $x^2+y…Free
  3. Q37Find the equation of the circle which passes through the points $(2,3)$ and $(4,5)$ and the centre lies on the straight line $y-4x+3=0$. (A)…Free
  4. Q38The equations of the tangents to the circle $x^2+y^2=4$ which are parallel to $x+2y+3=0$ are (A) $x-2y=2$ (B) $x+2y=\pm2\sqrt3$ (C) $x+2y=\p…Preview
  5. Q39If the lines $3x-4y+4=0$ and $6x-8y-7=0$ are tangents to a circle, then find the radius of the circle. (A) $\dfrac34$ (B) $\dfrac43$ (C) $\d…Preview
  6. Q40Area of the circle with centre at $(1,2)$ and passing through $(4,6)$ is (A) $5\pi$ (B) $10\pi$ (C) $25\pi$ (D) $100\pi$Preview
  7. Q41If a circle passes through the points $(0,0)$, $(a,0)$ and $(0,b)$ then find the co-ordinates of its centre. (A) $\left(-\dfrac a2,-\dfrac b…Preview
  8. Q42The equation of a circle with the origin as centre and passing through the vertices of an equilateral triangle whose median is of length $3a…Preview
  9. Q43A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at a point $A$, enclosing an angle of $…Preview
  10. Q44The parametric equations of the circle $x^2+y^2+mx+my=0$ are (A) $x=-\dfrac{m}{2}+\dfrac{m}{\sqrt2}\cos\theta,\ y=-\dfrac{m}{2}+\dfrac{m}{\s…Preview
+Show 31 questions31 questions
  1. Q45Find the centre and radius of the circle $x^2+y^2-x+2y-3=0$.Free
  2. Q46Find the centre and radius of the circle $x=3-4\sin\theta$, $y=2-4\cos\theta$.Free
  3. Q47Find the equation of the circle passing through the point of intersection of the lines $x+3y=0$ and $2x-7y=0$, whose centre is the point of…Free
  4. Q48Find the equation of the circle which passes through the origin and cuts off chords of length $4$ and $6$ on the positive sides of the X-axi…Preview
  5. Q49Show that the points $(9,1)$, $(7,9)$, $(-2,12)$ and $(6,10)$ are concyclic.Preview
  6. Q50The line $2x-y+6=0$ meets the circle $x^2+y^2+10x+9=0$ at $A$ and $B$. Find the equation of the circle on $AB$ as diameter.Preview
  7. Q51Show that $x=-1$ is a tangent to the circle $x^2+y^2-2y=0$ at the point $(-1,1)$.Preview
  8. Q52Find the equation of the tangent to the circle $x^2+y^2=64$ at the point $P\!\left(\dfrac{2\pi}{3}\right)$.Preview
  9. Q53Find the equation of the locus of the point of intersection of perpendicular tangents drawn to the circle $x=5\cos\theta$, $y=5\sin\theta$.Preview
  10. Q54Find the equation of the circle concentric with $x^2+y^2-4x+6y=1$ and having radius $4$ units.Preview
  11. Q55Find the lengths of the intercepts made on the co-ordinate axes by the circle $x^2+y^2-8x+y-20=0$.Preview
  12. Q56Find the lengths of the intercepts made on the co-ordinate axes by the circle $x^2+y^2-5x+13y-14=0$.Preview
  13. Q57Show that the circles $x^2+y^2-4x+10y+20=0$ and $x^2+y^2+8x-6y-24=0$ touch each other externally. Find their point of contact and the equati…Preview
  14. Q58Show that the circles $x^2+y^2-4x-10y+19=0$ and $x^2+y^2+2x+8y-23=0$ touch each other externally. Find their point of contact and the equati…Preview
  15. Q59Show that the circles $x^2+y^2-4x-4y-28=0$ and $x^2+y^2-4x-12=0$ touch each other internally. Find their point of contact and the equation o…Preview
  16. Q60Show that the circles $x^2+y^2+4x-12y+4=0$ and $x^2+y^2-2x-4y+4=0$ touch each other internally. Find their point of contact and the equation…Preview
  17. Q61Find the length of the tangent segment drawn from the point $(5,3)$ to the circle $x^2+y^2+10x-6y-17=0$.Preview
  18. Q62Find the value of $k$, if the length of the tangent segment from the point $(8,-3)$ to the circle $x^2+y^2-2x+ky-23=0$ is $10$.Preview
  19. Q63Find the equation of the tangent to the circle $x^2+y^2-6x-4y=0$ at the point $(6,4)$ on it.Preview
  20. Q64Find the equation of the tangent to the circle $x^2+y^2=5$ at the point $(1,-2)$ on it.Preview
  21. Q65Find the equation of the tangent to the circle $x=5\cos\theta$, $y=5\sin\theta$, at the point $\theta=\pi/3$ on it.Preview
  22. Q66Show that $2x+y+6=0$ is a tangent to $x^2+y^2+2x-2y-3=0$. Find its point of contact.Preview
  23. Q67If the tangent at $(3,-4)$ to the circle $x^2+y^2=25$ touches the circle $x^2+y^2+8x-4y+c=0$, find $c$.Preview
  24. Q68Find the equations of the tangents to the circle $x^2+y^2=16$ with slope $-2$.Preview
  25. Q69Find the equations of the tangents to the circle $x^2+y^2=4$ which are parallel to $3x+2y+1=0$.Preview
  26. Q70Find the equations of the tangents to the circle $x^2+y^2=36$ which are perpendicular to the line $5x+y=2$.Preview
  27. Q71Find the equations of the tangents to the circle $x^2+y^2-2x+8y-23=0$ having slope $3$.Preview
  28. Q72Find the equation of the locus of a point, the tangents from which to the circle $x^2+y^2=9$ are at right angles.Preview
  29. Q73Tangents to the circle $x^2+y^2=a^2$ with inclinations $\theta_1$ and $\theta_2$ intersect at $P$. Find the locus of $P$ such that $\tan\the…Preview
  30. Q74Tangents to the circle $x^2+y^2=a^2$ with inclinations $\theta_1$ and $\theta_2$ intersect at $P$. Find the locus of $P$ such that $\cot\the…Preview
  31. Q75Tangents to the circle $x^2+y^2=a^2$ with inclinations $\theta_1$ and $\theta_2$ intersect at $P$. Find the locus of $P$ such that $\cot\the…Preview