Mathematics · Textbook solutions

Circle

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 81 questions

6.1 Different Forms of the Equation of a Circle

5 q

Solved Examples

Worked · 5
  1. 6.1 SolvedEx.1
    Find the equation of a circle with centre at origin and radius 3.
  2. 6.1 SolvedEx.2
    Find the equation of a circle whose centre is (−3, 1)(-3,\ 1) and which pass through the point (5, 2)(5,\ 2).
  3. 6.1 SolvedEx.3
    Find the equation of the circle with A(2, −3)A(2,\ -3) and B(−3, 5)B(-3,\ 5) as end points of its diameter.
  4. 6.1 SolvedEx.4
    Find the equation of circle touching the Y-axis at point (0, 3)(0,\ 3) and whose Centre is at (−3, 3)(-3,\ 3).
  5. 6.1 SolvedEx.5
    Find the equation of the circle whose centre is at (3, −4)(3,\ -4) and the line 3x−4y−5=03x - 4y - 5 = 0 cuts the circle at A and B; where l(AB)=6l(AB) = 6.

Exercise 6.1

16 q
  1. Find the equation of the circle with
    Ex 6.1 Q1(i)
    Centre at origin and radius 4.
  2. Ex 6.1 Q1(ii)
    Centre at (−3, −2)(-3,\ -2) and radius 6.
  3. Ex 6.1 Q1(iii)
    Centre at (2, −3)(2,\ -3) and radius 5.
  4. Ex 6.1 Q1(iv)
    Centre at (−3, −3)(-3,\ -3) passing through point (−3, −6)(-3,\ -6)
  5. Find the centre and radius of the circle.
    Ex 6.1 Q2(i)
    x2+y2=25x^{2} + y^{2} = 25
  6. Ex 6.1 Q2(ii)
    (x−5)2+(y−3)2=20(x - 5)^{2} + (y - 3)^{2} = 20
  7. Ex 6.1 Q2(iii)
    (x−12)2+(y+13)2=136\left(x - \dfrac{1}{2}\right)^{2} + \left(y + \dfrac{1}{3}\right)^{2} = \dfrac{1}{36}
  8. Find the equation of the circle with centre
    Ex 6.1 Q3(i)
    At (a, b)(a,\ b) touching the Y-axis
  9. Ex 6.1 Q3(ii)
    At (−2, 3)(-2,\ 3) touching the X-axis
  10. Ex 6.1 Q3(iii)
    on the X-axis and passing through the origin having radius 4.
  11. Ex 6.1 Q3(iv)
    at (3, 1)(3,\ 1) and touching the line 8x−15y+25=08x - 15y + 25 = 0
  12. Ex 6.1 Q4
    Find the equation circle if the equations of two diameters are 2x+y=62x + y = 6 and 3x+2y=43x + 2y = 4. When radius of circle is 9.
  13. Ex 6.1 Q5
    If y=2xy = 2x is a chord of circle x2+y2−10x=0x^{2} + y^{2} - 10x = 0, find the equation of circle with this chord as diametre.
  14. Ex 6.1 Q6
    Find the equation of a circle with radius 4 units and touching both the co-ordinate axes having centre in third quadrant.
  15. Ex 6.1 Q7
    Find the equation of circle (a) passing through the origin and having intercepts 4 and −5-5 on the co-ordinate axes.
  16. Ex 6.1 Q8
    Find the equation of a circle passing through the points (1, −4)(1,\ -4), (5, 2)(5,\ 2) and having its centre on the line x−2y+9=0x - 2y + 9 = 0.

6.2 General Equation of a Circle

3 q

Solved Examples

Worked · 3
  1. 6.2 SolvedEx.1
    Prove that 3x2+3y2−6x+4y−1=03x^{2} + 3y^{2} - 6x + 4y - 1 = 0, represents a circle. Find its centre and radius.
  2. 6.2 SolvedEx.2
    Find the equation of the circle passing through the points (5, −6)(5,\ -6), (1, 2)(1,\ 2) and (3, −4)(3,\ -4).
  3. 6.2 SolvedEx.3
    Show that the points (5, 5)(5,\ 5), (6, 4)(6,\ 4), (−2, 4)(-2,\ 4) and (7, 1)(7,\ 1) are on the same circle; i.e. these points are concyclic.

Exercise 6.2

6 q
  1. Find the centre and radius of each of the following.
    Ex 6.2 Q1(i)
    x2+y2−2x+4y−4=0x^2 + y^2 - 2x + 4y - 4 = 0
  2. Ex 6.2 Q1(ii)
    x2+y2−6x−8y−24=0x^2 + y^2 - 6x - 8y - 24 = 0
  3. Ex 6.2 Q1(iii)
    4x2+4y2−24x−8y−24=04x^2 + 4y^2 - 24x - 8y - 24 = 0
  4. Ex 6.2 Q2
    Show that the equation 3x2+3y2+12x+18y−11=03x^2 + 3y^2 + 12x + 18y - 11 = 0 represents a circle.
  5. Ex 6.2 Q3
    Find the equation of the circle passing through the points (5,7)(5, 7), (6,6)(6, 6) and (2,−2)(2, -2).
  6. Ex 6.2 Q4
    Show that the points (3,−2)(3, -2), (1,0)(1, 0), (−1,−2)(-1, -2) and (1,−4)(1, -4) are concyclic.

6.3 Tangents and the Director Circle

3 q

Solved Examples

Worked · 3
  1. 6.3.4 SolvedEx.1
    Find the parametric equation of the circle x2+y2−6x+4y−3=0x^2 + y^2 - 6x + 4y - 3 = 0
  2. 6.3.4 SolvedEx.2
    Show that the line 3x−4y+15=03x - 4y + 15 = 0 is a tangent to the circle x2+y2=9x^2 + y^2 = 9. Find the point of contact.
  3. 6.3.4 SolvedEx.3
    Find the equation of the tangent to the circle x2+y2−4x−6y−12=0x^2 + y^2 - 4x - 6y - 12 = 0 at (−1,−1)(-1, -1)

Exercise 6.3

7 q
  1. Write the parametric equations of the circles
    Ex 6.3 Q1(i)
    x2+y2=9x^2 + y^2 = 9
  2. Ex 6.3 Q1(ii)
    x2+y2+2x−4y−4=0x^2 + y^2 + 2x - 4y - 4 = 0
  3. Ex 6.3 Q1(iii)
    (x−3)2+(y+4)2=25(x - 3)^2 + (y + 4)^2 = 25
  4. Ex 6.3 Q2
    Find the parametric representation of the circle 3x2+3y2−4x+6y−4=03x^2 + 3y^2 - 4x + 6y - 4 = 0.
  5. Ex 6.3 Q3
    Find the equation of a tangent to the circle x2+y2−3x+2y=0x^2 + y^2 - 3x + 2y = 0 at the origin.
  6. Ex 6.3 Q4
    Show that the line 7x−3y−1=07x - 3y - 1 = 0 touches the circle x2+y2+5x−7y+4=0x^2 + y^2 + 5x - 7y + 4 = 0 at point (1,2)(1, 2)
  7. Ex 6.3 Q5
    Find the equation of tangent to the circle x2+y2−4x+3y+2=0x^2 + y^2 - 4x + 3y + 2 = 0 at the point (4,−2)(4, -2)

Miscellaneous Exercise 6

41 q

(I) Choose the correct alternative

Practice · 10
  1. Misc I Q1
    Equation of a circle which passes through (3,6)(3, 6) and touches the axes is
    1. A.
      x2+y2+6x+6y+3=0x^2 + y^2 + 6x + 6y + 3 = 0
    2. B.
      x2+y2−6x−6y−9=0x^2 + y^2 - 6x - 6y - 9 = 0
    3. C.
      x2+y2−6x−6y+9=0x^2 + y^2 - 6x - 6y + 9 = 0
    4. D.
      x2+y2−6x+6y−3=0x^2 + y^2 - 6x + 6y - 3 = 0
  2. Misc I Q2
    If the lines 2x−3y=52x - 3y = 5 and 3x−4y=73x - 4y = 7 are the diameters of a circle of area 154 sq. units, then find the equation of the circle.
    1. A.
      x2+y2−2x+2y=40x^2 + y^2 - 2x + 2y = 40
    2. B.
      x2+y2−2x−2y=47x^2 + y^2 - 2x - 2y = 47
    3. C.
      x2+y2−2x+2y=47x^2 + y^2 - 2x + 2y = 47
    4. D.
      x2+y2−2x−2y=40x^2 + y^2 - 2x - 2y = 40
  3. Misc I Q3
    Find the equation of the circle which passes through the points (2,3)(2, 3) and (4,5)(4, 5) and the centre lies on the straight line y−4x+3=0y - 4x + 3 = 0.
    1. A.
      x2+y2−4x−10y+25=0x^2 + y^2 - 4x - 10y + 25 = 0
    2. B.
      x2+y2−4x−10y−25=0x^2 + y^2 - 4x - 10y - 25 = 0
    3. C.
      x2+y2−4x+10y−25=0x^2 + y^2 - 4x + 10y - 25 = 0
    4. D.
      x2+y2+4x−10y+25=0x^2 + y^2 + 4x - 10y + 25 = 0
  4. Misc I Q4
    The equation of the tangent to the circle x2+y2=4x^2 + y^2 = 4 which are parallel to x+2y+3=0x + 2y + 3 = 0 are
    1. A.
      x−2y=2x - 2y = 2
    2. B.
      x+2y=±23x + 2y = \pm 2\sqrt{3}
    3. C.
      x+2y=±25x + 2y = \pm 2\sqrt{5}
    4. D.
      x−2y=±25x - 2y = \pm 2\sqrt{5}
  5. Misc I Q5
    If the lines 3x−4y+4=03x - 4y + 4 = 0 and 6x−8y−7=06x - 8y - 7 = 0 are tangents to a circle, then find the radius of the circle
    1. A.
      34\frac{3}{4}
    2. B.
      43\frac{4}{3}
    3. C.
      14\frac{1}{4}
    4. D.
      74\frac{7}{4}
  6. Misc I Q6
    Area of the circle centre at (1,2)(1, 2) and passing through (4,6)(4, 6) is
    1. A.
      5π5\pi
    2. B.
      10π10\pi
    3. C.
      25π25\pi
    4. D.
      100π100\pi
  7. Misc I Q7
    If a circle passes through the point (0,0)(0, 0), (a,0)(a, 0) and (0,b)(0, b) then find the co-ordinates of its centre.
    1. A.
      (−a2,−b2)\left(\frac{-a}{2}, \frac{-b}{2}\right)
    2. B.
      (a2,−b2)\left(\frac{a}{2}, \frac{-b}{2}\right)
    3. C.
      (−a2,b2)\left(\frac{-a}{2}, \frac{b}{2}\right)
    4. D.
      (a2,b2)\left(\frac{a}{2}, \frac{b}{2}\right)
  8. Misc I Q8
    The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a3a is
    1. A.
      x2+y2=9a2x^2 + y^2 = 9a^2
    2. B.
      x2+y2=16a2x^2 + y^2 = 16a^2
    3. C.
      x2+y2=4a2x^2 + y^2 = 4a^2
    4. D.
      x2+y2=a2x^2 + y^2 = a^2
  9. Misc I Q9
    A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A enclosing an angle of 60∘60^{\circ}. The area enclosed by these tangents and the area of the circle is
    1. A.
      23−π6\frac{2}{\sqrt{3}} - \frac{\pi}{6}
    2. B.
      3−π3\sqrt{3} - \frac{\pi}{3}
    3. C.
      π3−36\frac{\pi}{3} - \frac{\sqrt{3}}{6}
    4. D.
      3(1−π6)\sqrt{3}\left(1 - \frac{\pi}{6}\right)
  10. Misc I Q10
    The parametric equations of the circle x2+y2+mx+my=0x^2 + y^2 + mx + my = 0 are
    1. A.
      x=−m2+m2cos⁡θ, y=−m2+m2sin⁡θx = \frac{-m}{2} + \frac{m}{\sqrt{2}}\cos\theta,\ y = \frac{-m}{2} + \frac{m}{\sqrt{2}}\sin\theta
    2. B.
      x=−m2+m2cos⁡θ, y=+m2+m2sin⁡θx = \frac{-m}{2} + \frac{m}{\sqrt{2}}\cos\theta,\ y = \frac{+m}{2} + \frac{m}{\sqrt{2}}\sin\theta
    3. C.
      x=0, y=0x = 0,\ y = 0
    4. D.
      x=mcos⁡θ ; y=msin⁡θx = m\cos\theta\ ;\ y = m\sin\theta

(II) Answer the following

Practice · 31
  1. Misc II Q1
    Find the centre and radius of the circle x2+y2−x+2y−3=0x^2 + y^2 - x + 2y - 3 = 0
  2. Misc II Q2
    Find the centre and radius of the circle x=3−4sin⁡θx = 3 - 4\sin\theta, y=2−4cos⁡θy = 2 - 4\cos\theta
  3. Misc II Q3
    Find the equation of circle passing through the point of intersection of the lines x+3y=0x + 3y = 0 and 2x−7y=02x - 7y = 0 whose centre is the point of intersection of lines x+y+1=0x + y + 1 = 0 and x−2y+4=0x - 2y + 4 = 0.
  4. Misc II Q4
    Find the equation of circle which passes through the origin and cuts off chords of length 4 and 6 on the positive side of xx-axis and yy-axis respectively.
  5. Misc II Q5
    Show that the points (9,1)(9, 1), (7,9)(7, 9), (−2,12)(-2, 12) and (6,10)(6, 10) are concyclic.
  6. Misc II Q6
    The line 2x−y+6=02x - y + 6 = 0 meets the circle x2+y2+10x+9=0x^2 + y^2 + 10x + 9 = 0 at A and B. Find the equation of circle on AB as diameter.
  7. Misc II Q7
    Show that x=−1x = -1 is a tangent to circle x2+y2−2y=0x^2 + y^2 - 2y = 0 at (−1,1)(-1, 1).
  8. Misc II Q8
    Find the equation of tangent to the circle x2+y2=64x^2 + y^2 = 64 at the point P(2π3)P\left(\frac{2\pi}{3}\right)
  9. Misc II Q9
    Find the equation of locus of the point of intersection of perpendicular tangents drawn to the circle x=5cos⁡θx = 5\cos\theta and y=5sin⁡θy = 5\sin\theta.
  10. Misc II Q10
    Find the equation of the circle concentric with x2+y2−4x+6y=1x^2 + y^2 - 4x + 6y = 1 and having radius 4 units.
  11. Find the lengths of the intercepts made on the co-ordinate axes, by the circle.
    Misc II Q11(i)
    x2+y2−8x+y−20=0x^2 + y^2 - 8x + y - 20 = 0
  12. Misc II Q11(ii)
    x2+y2−5x+13y−14=0x^2 + y^2 - 5x + 13y - 14 = 0
  13. Show that the circles touch each other externally. Find their point of contact and the equation of their common tangent.
    Misc II Q12(i)
    x2+y2−4x+10y+20=0x^2 + y^2 - 4x + 10y + 20 = 0, x2+y2+8x−6y−24=0x^2 + y^2 + 8x - 6y - 24 = 0.
  14. Misc II Q12(ii)
    x2+y2−4x−10y+19=0x^2 + y^2 - 4x - 10y + 19 = 0, x2+y2+2x+8y−23=0x^2 + y^2 + 2x + 8y - 23 = 0.
  15. Show that the circles touch each other internally. Find their point of contact and the equation of their common tangent.
    Misc II Q13(i)
    x2+y2−4x−4y−28=0x^2 + y^2 - 4x - 4y - 28 = 0, x2+y2−4x−12=0x^2 + y^2 - 4x - 12 = 0.
  16. Misc II Q13(ii)
    x2+y2+4x−12y+4=0x^2 + y^2 + 4x - 12y + 4 = 0, x2+y2−2x−4y+4=0x^2 + y^2 - 2x - 4y + 4 = 0.
  17. Misc II Q14
    Find the length of the tangent segment drawn from the point (5,3)(5, 3) to the circle x2+y2+10x−6y−17=0x^2 + y^2 + 10x - 6y - 17 = 0.
  18. Misc II Q15
    Find the value of kk, if the length of the tangent segment from the point (8,−3)(8, -3) to the circle x2+y2−2x+ky−23=0x^2 + y^2 - 2x + ky - 23 = 0 is 10\sqrt{10}.
  19. Misc II Q16
    Find the equation of tangent to Circle x2+y2−6x−4y=0x^2 + y^2 - 6x - 4y = 0, at the point (6,4)(6, 4) on it.
  20. Misc II Q17
    Find the equation of tangent to Circle x2+y2=5x^2 + y^2 = 5, at the point (1,−2)(1, -2) on it.
  21. Misc II Q18
    Find the equation of tangent to Circle x=5cos⁡θx = 5\cos\theta, y=5sin⁡θy = 5\sin\theta, at the point θ=π3\theta = \frac{\pi}{3} on it.
  22. Misc II Q19
    Show that 2x+y+6=02x + y + 6 = 0 is a tangent to x2+y2+2x−2y−3=0x^2 + y^2 + 2x - 2y - 3 = 0. Find its point of contact.
  23. Misc II Q20
    If the tangent at (3,−4)(3, -4) to the circle x2+y2=25x^2 + y^2 = 25 touches the circle x2+y2+8x−4y+c=0x^2 + y^2 + 8x - 4y + c = 0, find cc.
  24. Misc II Q21
    Find the equations of the tangents to the circle x2+y2=16x^2 + y^2 = 16 with slope −2-2.
  25. Misc II Q22
    Find the equations of the tangents to the circle x2+y2=4x^2 + y^2 = 4 which are parallel to 3x+2y+1=03x + 2y + 1 = 0.
  26. Misc II Q23
    Find the equations of the tangents to the circle x2+y2=36x^2 + y^2 = 36 which are perpendicular to the line 5x+y=25x + y = 2.
  27. Misc II Q24
    Find the equations of the tangents to the circle x2+y2−2x+8y−23=0x^2 + y^2 - 2x + 8y - 23 = 0 having slope 3.
  28. Misc II Q25
    Find the equation of the locus of a point, the tangents from which to the circle x2+y2=9x^2 + y^2 = 9 are at right angles.
  29. Tangents to the circle x2+y2=a2x^2 + y^2 = a^2 with inclinations, θ1\theta_1 and θ2\theta_2 intersect in P. Find the locus of such that
    Misc II Q26(i)
    tan⁡θ1+tan⁡θ2=0\tan\theta_1 + \tan\theta_2 = 0
  30. Misc II Q26(ii)
    cot⁡θ1+cot⁡θ2=5\cot\theta_1 + \cot\theta_2 = 5
  31. Misc II Q26(iii)
    cot⁡θ1⋅cot⁡θ2=c\cot\theta_1 \cdot \cot\theta_2 = c