![Questions (4)
1. Find the equation to the straight line which passes through the
point ሺ
5, 6ሻand intercepts on the axes
[1]. Equal in magnitude and both positive,
[2]. Equal in magnitude but opposite in sign.
2. Find the equation to the straight lines which pass through the
point ሺ
1, −2ሻand cut off equal distances from the two axes.
3. Find the equation to the straight line which passes through the
given point ሺ
𝑥′
, 𝑦
′ሻand is such that the given point bisects the
part intercepted between the axes.
4. Find the equation to the straight line which passes through the
point ሺ
−4, 3ሻ and is such that the portion of it between the axes
is divided by the point in the ratio 5 ∶ 3.](https://image.slidesharecdn.com/coordinategeometry-241015090813-d57bc9fc/85/Coordinate-Geometry-two-dimensions-mathematics-54-320.jpg)
![POINT OF INTERSECTION
Find the coordinates of the points of intersection of the straight lines
whose equations are
1. 2𝑥− 3𝑦+ 5 = 0𝑎𝑛𝑑7𝑥+ 4𝑦= 3.
2.
𝑥
𝑎
+
𝑦
𝑏
= 1𝑎𝑛𝑑
𝑥
𝑏
+
𝑦
𝑎
= 1.
3. 2𝑥− 3𝑦= 1𝑎𝑛𝑑5𝑦− 𝑥= 3,and the angle at which they cut
one another.
4. 3𝑥+ 𝑦+ 12 = 0𝑎𝑛𝑑𝑥+ 2𝑦− 1 = 0and the angle at which
they cut one another.
5. Prove that the following sets of three lines meet in a point.
[1]. 2𝑥− 3𝑦= 7,3𝑥− 4𝑦= 13,𝑎𝑛𝑑8𝑥− 11𝑦= 33
[2]. 3𝑥+ 4𝑦+ 6 = 0,6𝑥+ 5𝑦= −9,𝑎𝑛𝑑3𝑥+ 3𝑦= −5.](https://image.slidesharecdn.com/coordinategeometry-241015090813-d57bc9fc/85/Coordinate-Geometry-two-dimensions-mathematics-62-320.jpg)
Coordinate Geometry two dimensions mathematics
- 1. Coordinate Geometry by S. L. Loney
- 2. Geometry is concerned with properties of space that are related to distance, shape, size, and the relative position of figures. Algebra is the study of mathematical symbols and the rules for manipulating these symbols appearing in equations describing relationships between variables. René Descartes coupled the branch of geometry with that of algebra, and thereby the study of geometry could be simplified to a very large extent. Of course, this hindered the development of abstract geometry, just like the availability of digital computers blocked the growth of mathematics.
- 29. Questions (10) Find the distance between the following pairs of points. 1. ሺ 2,3ሻ𝑎𝑛𝑑ሺ 5,7ሻ 2. ሺ 1,−7ሻ𝑎𝑛𝑑ሺ −1,5ሻ 3. ሺ −3,−2ሻand ሺ −6,7ሻ 4. ሺ 𝑎,0ሻ𝑎𝑛𝑑ሺ 0,𝑏ሻ 5. ሺ 𝑏+ 𝑐, 𝑐+ 𝑎ሻ𝑎𝑛𝑑ሺ 𝑐+ 𝑎, 𝑎+ 𝑏ሻ 6. ሺ 𝑎cos𝛼,𝑎sin𝛼ሻ𝑎𝑛𝑑ሺ 𝑎cos𝛽, 𝑎sin𝛽ሻ 7. ሺ 𝑎𝑚1 2 , 2𝑎𝑚1ሻ𝑎𝑛𝑑ሺ 𝑎𝑚2 2 ,2𝑎𝑚2ሻ 8. Lay down in a figure the positions of the points ሺ 1,−3ሻand ሺ −2,1ሻand prove that the distance between them is 5. 9. Find the value of 𝑥 1 if the distance between the points ሺ 𝑥 1, 2ሻ𝑎𝑛𝑑ሺ 3, 4ሻ be 8. 10.A line is of length 10 and one end is at the point ሺ 2,−3ሻ ; if the abscissa of the other end be 10, prove that its ordinate must be 3 or -9.
- 30. Questions (8) 1. Prove that the points ሺ 2𝑎, 4𝑎ሻ,ሺ 2𝑎, 6𝑎ሻ, and ൫ 2𝑎+ ξ3𝑎, 5𝑎൯ are the vertices of an equilateral triangle whose side is 2a. 2. Prove that the points ሺ −2, −1ሻ,ሺ 1, 0ሻ,ሺ 4, 3ሻ ,𝑎𝑛𝑑 ሺ 1, 2ሻ are at the vertices of a parallelogram. 3. Prove that the points ሺ 2, −2ሻ,ሺ 8, 4ሻ , ሺ 5, 7ሻ,𝑎𝑛𝑑 ሺ −1, 1ሻare at the angular points of a rectangle. 4. Prove that the points ቀ− 1 14 , 39 14 ቁ is the centre of the circle circumscribing the triangle whose angular points are ሺ 1, 1ሻ ,ሺ 2, 3ሻ , 𝑎𝑛𝑑 ሺ −2, 2ሻ . 5. Find the coordinates of the point which divides the line joining the points ሺ 1, 3ሻ 𝑎𝑛𝑑 ሺ 2, 7ሻin the ratio 3:4. 6. Find the coordinates of the point which divides the same line in the ration 3: − 4. 7. Find the coordinates of the point which divides, internally and externally, the line joining ሺ −1, 2ሻto ሺ 4, 5ሻ in the ratio 2:3. 8. Find the coordinates of the point which divides, internally and externally, the line joining ሺ −3, −4ሻ 𝑡𝑜 ሺ −8, 7ሻin the ratio 7:5.
- 31. Questions (3) 1. Thelinejoiningthepoints ሺ 1, −2ሻand ሺ −3, 4ሻistrisected;findthe coordinatesofthepointsoftrisection. 2. Thelinejoiningthepoints ሺ −6, 8ሻ𝑎𝑛𝑑ሺ 8, −6ሻisdividedintofourequal parts;findthe coordinates ofthepointsofsection. 3. Findthecoordinates ofthepointswhich divide,internallyandexternally, thelinejoiningthepoint ሺ 𝑎+𝑏, 𝑎−𝑏ሻtothepoint ሺ 𝑎−𝑏, 𝑎+𝑏ሻinthe ratio𝑎:𝑏.
- 35. Questions (9) Find the area of the triangles the coordinates of whose angular points are respectively 1. ሺ 1, 3ሻ,ሺ −7, 6ሻ 𝑎𝑛𝑑 ሺ 5, −1ሻ 2. ሺ 0, 4ሻ, ሺ 3, 6ሻ 𝑎𝑛𝑑 ሺ −8, −2ሻ 3. ሺ 5, 2ሻ, ሺ −9, −3ሻ 𝑎𝑛𝑑ሺ −3, −5ሻ 4. ሺ 𝑎, 𝑏+ 𝑐ሻ, ሺ 𝑎, 𝑏− 𝑐ሻ 𝑎𝑛𝑑൫ –𝑎, 𝑐൯ 5. ሺ 𝑎, 𝑐+ 𝑎ሻ, ሺ 𝑎, 𝑐ሻ 𝑎𝑛𝑑 ሺ −𝑎, 𝑐− 𝑎ሻ 6. ሺ 𝑎cos𝜙1, 𝑏sin𝜙1ሻ, ሺ 𝑎cos𝜙2, 𝑏sin𝜙2ሻ 𝑎𝑛𝑑ሺ 𝑎cos𝜙3, 𝑏sin𝜙3ሻ 7. ሺ 𝑎𝑚1 2 , 2𝑎𝑚1ሻ,ሺ 𝑎𝑚2 2 , 2𝑎𝑚2ሻ 𝑎𝑛𝑑 ሺ 𝑎𝑚3 2 , 2𝑎𝑚3ሻ 8. ሼ 𝑎𝑚1𝑚2, 𝑎ሺ 𝑚1 + 𝑚2ሻሽ ,ሼ 𝑎𝑚2𝑚3, 𝑎ሺ 𝑚2 + 𝑚3ሻሽ and ሼ 𝑎𝑚3𝑚1, 𝑎ሺ 𝑚3 + 𝑚1ሻሽ 9. ቄ 𝑎𝑚1, 𝑎 𝑚1 ቅ, ቄ 𝑎𝑚2, 𝑎 𝑚2 ቅ, 𝑎𝑛𝑑 ቄ 𝑎𝑚3, 𝑎 𝑚3 ቅ
- 36. Questions (3) Prove(byshowingthattheareaofthetriangleformedbythemiszero)that thefollowingsetsofthreepointsareinaStraightline: 1. ሺ1, 4ሻ, ሺ3, −2ሻ, 𝑎𝑛𝑑 ሺ−3, 16ሻ 2. ቀ− 1 2 , 3ቁ, ሺ−5, 6ሻ, 𝑎𝑛𝑑 ሺ−8, 8ሻ 3. ሺ𝑎, 𝑏+𝑐ሻ, ሺ𝑏, 𝑐+𝑎ሻ, 𝑎𝑛𝑑 ሺ𝑐, 𝑎+𝑏ሻ
- 37. Locus. Equation to a Locus • Article 36: When a point moves so as to satisfy a given condition or conditions, the path it traces out is called its locus under these conditions. • For example, suppose O is a given point in the plane of the paper and that a point P is to move on the paper so that its distance from O is constant and equal to a. It is clear that all the positions of the moving point must lie on the circumference of a circle whose center is O and radius is a. The circumference of this circle is therefore the "locus" of P when it moves, subject to the condition that its distance from O shall be equal to the constant distance a.
- 38. Equation to a curve • Definition The equation for a curve is the relation that exists between the coordinates of any point on the curve and that holds for no other points except those lying on the curve.
- 39. Straight Line Article 46 To find the equation for a straight line that is parallel to one of the coordinate axes. Let CL be any line parallel to the axis of y and passing through a point C on the axis of x such that OC = c. Let P be any point on this line whose coordinates are x and y. Then the abscissa of the point P is always c, so that x=c ……..(1)
- 40. Straight line parallel to the axes Y L P O C
- 41. Continued The expression (1) is true for every point on the line CL (produced indefinitely both ways), and for no other point is, by article 42, the equation to the line. It should be noted that the equation does not contain the coordinate y. Similarly, the equation for a straight line parallel to the axis of x is y = d. Corollary: The equation to the axis of x is y=0.The equation for the y axis is x = 0.
- 43. 47. To find the equation to a straight line which cuts off a given intercept on the axis y and is inclined at a given angle to the axis of x. Y L P C N L’ O M X
- 44. Let the given intercept be c and let the given angle be 𝛼. Let C be a point on the axis of y such that OC is c. Through C draw a straight line LCL’ inclined at an angle 𝛼ሺ = tan−1 𝑚ሻto the axis of x, so that tan𝛼= 𝑚. The straight line LCL’ is therefore the straight line required, and we have to find the relation between the coordinates of any point P lying on it. Draw PM perpendicular to OX to meet in N a line through C parallel to OX. Let the coordinates of P be x and y, so that 𝑂𝑀= 𝑥𝑎𝑛𝑑𝑀 𝑃= 𝑦. Then 𝑀 𝑃= 𝑁 𝑃+ 𝑀 𝑁= 𝐶 𝑁tan𝛼+ 𝑂𝐶= 𝑚𝑥+ 𝑐, i.e. 𝑦= 𝑚𝑥+ 𝑐 This relation being true for any point on the given straight line is, by Art 42, the equation to the straight line. Corollary: The equation to nay straight line passing through the origin, i.e. which cuts off a zero intercept from the axis of y, is found by putting 𝑐= 0 and hence is 𝑦= 𝑚𝑥.
- 47. 50. To find the equation to the straight line which cuts off given intercepts a and b from the axes. Y B P O M A X
- 48. Let A and B be on OX and OY respectively, and be such that 𝑂𝐴= 𝑎𝑎𝑛𝑑𝑂𝐵= 𝑏. Join AB and produce it indefinitely both ways. Let P be any point ሺ 𝑥, 𝑦ሻon this straight line, and draw PM perpendicular to OX. We require the relation that always hold between 𝑥and 𝑦, so long as P lies on AB. By geometry, we have ሺ 𝑠𝑖𝑛𝑐𝑒∆𝐵𝑂𝐴𝑎𝑛𝑑∆𝑃𝑀 𝐴𝑎𝑟𝑒𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠ሻ 𝑂𝑀 𝑂𝐴 = 𝑃𝐵 𝐴 𝐵 ,𝑎𝑛𝑑 𝑀 𝑃 𝑂𝐵 = 𝐴 𝑃 𝐴 𝐵 𝑂𝑀 𝑂𝐴 + 𝑀 𝑃 𝑂𝐵 = 𝑃𝐵+𝐴 𝑃 𝐴 𝐵 = 1 i.e. 𝑥 𝑎 + 𝑦 𝑏 = 1 This is therefore the required equation; for it is the relation that holds between the coordinates of any point lying on the given straight line.
- 49. 51. Example: Find the equation to the straight line passing through the point ሺ 3, −4ሻand cutting off intercepts, equal but of opposite signs, from the axes. The equation to the straight line is then 𝑥 𝑎 + 𝑦 −𝑎 = 1 i.e. 𝑥− 𝑦= 𝑎 … (1) Since, in addition, the straight is to go through the point ሺ 3, −4ሻ, these coordinates must satisfy (1), so that 3− ሺ −4ሻ= 𝑎 and therefore 𝑎 = 7. The required equation is therefore 𝑥− 𝑦= 7.
- 50. 62. To find the equation to the straight line which passes through the two given points (x’, y’) and (x’’, y’’). Y (x’’, y’’) (x’, y’) O X
- 51. By Art. 47, the equation to any straight line is 𝑦= 𝑚𝑥+𝑐 ….(1) By properly determining the quantities 𝑚𝑎𝑛𝑑𝑐we can make (1) represent nay straight line we need. If (1) pass through the point ሺ 𝑥′ , 𝑦′ሻ, we have 𝑦′ = 𝑚𝑥′ +𝑐 ……(2) Substituting for 𝑐, from (2), the equation (1) becomes 𝑦−𝑦′ = 𝑚ሺ 𝑥−𝑥′ሻ ……(3)
- 52. This is the equation to the line going through ሺ 𝑥′ , 𝑦′ሻmaking an angle tan−1 𝑚with OX. If in addition (3) passes through the point ሺ 𝑥′′ , 𝑦′′ሻ, then 𝑦′′ −𝑦′ = 𝑚ሺ 𝑥′′ −𝑥′ሻ This yields 𝑚= 𝑦′′−𝑦′ 𝑥′′ −𝑥′ . Substituting this value in (3), we get as the required equation 𝑦−𝑦′ = ሺ 𝑦′′ −𝑦′ሻ ሺ 𝑥′′ −𝑥′ሻ ሼ 𝑥−𝑥′ሽ
- 53. Questions (6) Find the equation to the straight line 1. Cutting off an intercept unity from the positive direction of the axis of 𝑦and inclined at 45°to the axis of 𝑥. 2. Cutting off an intercept −5from the axis of 𝑦and being equally inclined to the axes. 3. Cutting off an intercept 2from the negative direction of the axis of 𝑦and inclined at 30°to OX. 4. Cutting of an intercept −3from the axis of 𝑦and inclined at an angle tan−1 3 5 to the axis of 𝑥. 5. Cutting of intercepts 3𝑎𝑛𝑑2from the axes. 6. Cutting of intercepts −5𝑎𝑛𝑑6from the axes.
- 54. Questions (4) 1. Find the equation to the straight line which passes through the point ሺ 5, 6ሻand intercepts on the axes [1]. Equal in magnitude and both positive, [2]. Equal in magnitude but opposite in sign. 2. Find the equation to the straight lines which pass through the point ሺ 1, −2ሻand cut off equal distances from the two axes. 3. Find the equation to the straight line which passes through the given point ሺ 𝑥′ , 𝑦 ′ሻand is such that the given point bisects the part intercepted between the axes. 4. Find the equation to the straight line which passes through the point ሺ −4, 3ሻ and is such that the portion of it between the axes is divided by the point in the ratio 5 ∶ 3.
- 55. Questions (7) Find the equations to the straight lines passing through the following pairs of points. 1. ሺ 0, 0ሻ𝑎𝑛𝑑ሺ 2, −2ሻ 2. ሺ 3, 4ሻ𝑎𝑛𝑑ሺ 5, 6ሻ 3. ሺ −1, 3ሻ 𝑎𝑛𝑑ሺ 6, −7ሻ 4. ሺ 0, 𝑎ሻ𝑎𝑛𝑑ሺ 𝑏, 0ሻ 5. ሺ 𝑎, 𝑏ሻ𝑎𝑛𝑑ሺ 𝑎+𝑏, 𝑎− 𝑏ሻ 6. ሺ 𝑎𝑡1 2 , 2𝑎𝑡1ሻ𝑎𝑛𝑑ሺ 𝑎𝑡2 2 , 2𝑎𝑡2ሻ 7. ቀ𝑎𝑡1, 𝑎 𝑡1 ቁ 𝑎𝑛𝑑 ቀ𝑎𝑡2, 𝑎 𝑡2 ቁ
- 56. Questions (6) Find the equations to the straight lines passing through the following pairs of points. 1. ሺ 𝑎cos𝜙1, 𝑎sin𝜙1ሻ𝑎𝑛𝑑ሺ 𝑎cos𝜙2, 𝑎sin𝜙2ሻ 2. ሺ 𝑎cos𝜙1, 𝑏sin𝜙1ሻ𝑎𝑛𝑑ሺ 𝑎cos𝜙2, 𝑏sin𝜙2ሻ 3. ሺ 𝑎sec𝜙1, 𝑏tan𝜙1ሻ𝑎𝑛𝑑ሺ 𝑎sec𝜙2, 𝑏tan𝜙2ሻ Find the equations to the sides of the triangles the coordinates of whose angular points are respectively. 4. ሺ 1, 4ሻ, ሺ 2, −3ሻ,𝑎𝑛𝑑ሺ −1, −2ሻ 5. ሺ 0, 1ሻ, ሺ 2, 0ሻ,𝑎𝑛𝑑ሺ −1, −2ሻ 6. Find the equations to the diagonals of the rectangle the equations of whose sides are 𝑥= 𝑎,𝑥= 𝑎′ ,𝑦= 𝑏,𝑎𝑛𝑑𝑦= 𝑏′.
- 57. 66. To find the angle between two given straight lines Y A C2 C1 L2 L1 O X
- 58. Let the two straight lines be AL1 and AL2, meeting the axes of 𝑥𝑖𝑛𝐿 1 and L2. Let their equations be 𝑦= 𝑚1𝑥+ 𝑐 1 𝑎𝑛𝑑 𝑦= 𝑚2𝑥+ 𝑐2 ……….(1) We know that tan𝐴 𝐿 1𝑋= 𝑚1,𝑎𝑛𝑑 tan𝐴 𝐿 2𝑋= 𝑚2 Now ∠ 𝐿 1𝐴 𝐿 2 = ∠ 𝐴 𝐿 1𝑋− ∠ 𝐴 𝐿 2𝑋 . ∴ tan∠ 𝐿 1𝐴 𝐿 2 = tanሾ ∠ 𝐴 𝐿 1𝑋− ∠ 𝐴 𝐿 2𝑋 ሿ tan 𝐴𝐿1𝑋 −tan 𝐴𝐿2𝑋 1+tan 𝐴 𝐿1𝑋 ∙tan 𝐴𝐿2𝑋 = 𝑚1−𝑚2 1+𝑚1𝑚2 Hence the required angle ∠ 𝐿 1𝐴 𝐿 2 = tan−1 𝑚1−𝑚2 1+𝑚1𝑚2 ….(2)
- 59. 67. To find the condition that two straight lines may be parallel. Two straight lines are parallel when the angle between them is zero and therefore the tangent of this angle is zero. This gives tan0°= 0 = 𝑚1 − 𝑚2 1+ 𝑚1𝑚2 ; 𝑚1 = 𝑚2 Two straight lines having same 𝑚will be parallel. 69. To find the condition that two straight lines may be perpendicular. tan90°= ∞ = 𝑚1 − 𝑚2 1+ 𝑚1𝑚2 ; 1+ 𝑚1 ∙𝑚2 = 0; 𝑚1 ∙𝑚2 = −1 The straight line 𝑦= 𝑚1𝑥+ 𝑐 1 is therefore perpendicular to 𝑦= 𝑚2𝑥+ 𝑐 2, if 𝑚1 = − 1 𝑚2 .
- 60. Questions (6) Find the angles between the pairs of straight lines 1. 𝑥− 𝑦ξ3 = 5𝑎𝑛𝑑ξ3𝑥+ 𝑦= 7. 2. 𝑥− 4𝑦= 3𝑎𝑛𝑑6𝑥− 𝑦= 11. 3. 𝑦= 3𝑥+ 7𝑎𝑛𝑑3𝑦− 𝑥= 8 4. 𝑦= ൫ 2− √3൯ 𝑥+ 5𝑎𝑛𝑑𝑦= ൫ 2+ √3൯ 𝑥− 7. 5. Find the tangent of the angle between the lines whose intercepts on the axes are respectively 𝑎,−𝑏𝑎𝑛𝑑𝑏,−𝑎. 6. Prove that the points ሺ 2, −1ሻ,ሺ 0, 2ሻ,ሺ 2, 3ሻ,𝑎𝑛𝑑ሺ 4, 0ሻ are the coordinates of the angular points of a parallelogram and find the angle between its diagonal.
- 61. Questions (4) Find the equation to the straight line 1. passing through the point ሺ 2, 3ሻand perpendicular to the straight line 4𝑥−3𝑦= 10. 2. passing through the point ሺ −6, 10ሻand perpendicular to the straight line 7𝑥+8𝑦= 5. 3. passing through the point ሺ 2, −3ሻand perpendicular to the straight line joining the points ሺ 5, 7ሻ𝑎𝑛𝑑ሺ −6, 3ሻ. 4. passing through the point ሺ −4, −3ሻand perpendicular to the straight line joining the points ሺ 1, 3ሻ𝑎𝑛𝑑ሺ 2, 7ሻ.
- 62. POINT OF INTERSECTION Find the coordinates of the points of intersection of the straight lines whose equations are 1. 2𝑥− 3𝑦+ 5 = 0𝑎𝑛𝑑7𝑥+ 4𝑦= 3. 2. 𝑥 𝑎 + 𝑦 𝑏 = 1𝑎𝑛𝑑 𝑥 𝑏 + 𝑦 𝑎 = 1. 3. 2𝑥− 3𝑦= 1𝑎𝑛𝑑5𝑦− 𝑥= 3,and the angle at which they cut one another. 4. 3𝑥+ 𝑦+ 12 = 0𝑎𝑛𝑑𝑥+ 2𝑦− 1 = 0and the angle at which they cut one another. 5. Prove that the following sets of three lines meet in a point. [1]. 2𝑥− 3𝑦= 7,3𝑥− 4𝑦= 13,𝑎𝑛𝑑8𝑥− 11𝑦= 33 [2]. 3𝑥+ 4𝑦+ 6 = 0,6𝑥+ 5𝑦= −9,𝑎𝑛𝑑3𝑥+ 3𝑦= −5.
- 63. The circle: Def. A circle is the locus of a point which moves so that its distance from a fixed point, called the centre, is equal to a given distance. The given distance is called the radius of the circle. Y P X M O O M
- 64. 139. To find the equation to a circle, having its centre at the origin. Let O be the centre of the circle and let 𝑎be its radius. Let OX and OY be the axes of coordinates. Let P be any point on the circumference of the circle, and its coordinates be 𝑥𝑎𝑛𝑑𝑦. Draw PM perpendicular to OX and join OP. Then 𝑂𝑀 2 +𝑀 𝑃2 = 𝑎2 i.e. 𝑥2 +𝑦2 = 𝑎2 This being the relation which hold between the coordinates of any point on the circumference is the required equation of the circle.
- 65. 140. To find the equation to a circle referred to any rectangular axes. Y P O M N X C L
- 66. Let OX and OY be the two rectangular axes. Let C be the centre of the circle and 𝑎its radius. Take any point P on the circumference and draw perpendicular CM and PN upon OX; Let P be the point ሺ 𝑥, 𝑦ሻ. Let the coordinates of C be ℎ𝑎𝑛𝑑𝑘; these are supposed to be known. We have 𝐶 𝐿= 𝑀 𝑁= 𝑂𝑁− 𝑂𝑀= 𝑥− ℎ, And 𝐿 𝑃= 𝑁 𝑃− 𝑁 𝐿= 𝑁 𝑃− 𝑀 𝐶= 𝑦− 𝑘. Hence, since 𝐶 𝐿 2 + 𝐿 𝑃2 = 𝐶 𝑃2 , We have ሺ 𝑥− ℎሻ2 + ሺ 𝑦− 𝑘ሻ2 = 𝑎2 This is the required equation.
- 68. Questions (7) Find the equation to the circle 1. Whose radius is 3and whose centre is ሺ −1, 2ሻ . 2. Whose radius is 10 and whose centre is ሺ −5, −6ሻ . 3. Whose radius is 𝑎 + 𝑏and whose centre is ሺ 𝑎, −𝑏ሻ . Find the coordinates of the centers and the radii of the circles whose equations are 1. 𝑥2 + 𝑦2 − 4𝑥− 8𝑦= 41 2. 3𝑥2 + 3𝑦2 − 5𝑥− 6𝑦+ 4 = 0 Find the equations to the circles which pass through the points 1. ሺ 1, 2ሻ ,ሺ 3, −4ሻ ,𝑎𝑛𝑑ሺ 5, −6ሻ 2. ሺ 1, 1ሻ ,ሺ 2, −1ሻ ,𝑎𝑛𝑑ሺ 3, 2ሻ
- 69. Acknowledgment The author acknowledges all the websites that helped a lot in preparing the slides presented here meant for B.Sc. (Ag) students. D C Agrawal dca_bhu@yahoo.com Note: There are a couple of typographical errors; you are supposed to find them out.
Editor's Notes
- #61: Examples VI; page 48