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- 1. POINT & STRAIGHT LINES
- 2. DISTANCE BETWEEN TWO POINTS IN A PLANE If A(x1, y1) and B (x2, y2) be two points in a plane, then distance between them is : 2 2 AB x2 x1 y2 y1 Distance of O (0, 0) from P (x, y) is OP x 2 y2.
- 3. EXAMPLES Show that four points (0, –1), (6, 7) (–2, 3) and (8, 3) are the vertices of a rectangle. Find the coordinates of the circumcenter of the triangle whose vertices are (8, 6), (8, –2) and (2, –2). Also find its circumradius. Ans. (5, 2), 5
- 4. POINT OF DIVISION ON A LINE SEGMENT Internal Division Let [AB] be the line segment joining two points A (x1, y1) and B (x2, y2) and let P (x, y) be any point on [AB] between A and B such that . m1 AP Then : PB m 2 m1x 2 m2 x1 m1 y 2 m 2 y1 x ,y , m1 m2 0 m1 m2 m1 m2
- 5. External Division If P divides the line segment [AB] externally in the ratio m1 : m2, then its co-ordinates are : m1 x 2 m2 x1 m1 y2 m2 y1 x ,y , m1 m2 m1 m2 m1 m2 Cor. Mid-point Formula : The co-ordinates of the mid- point of the join of (x1, y1) and (x2, y2) are : x1 x 2 y1 y 2 , 2 2
- 6. EXAMPLES In what ratio does the point (–1, –1) divide the line segment joining the points (4, 4) and (7, 7)? Ans. 5 : 8 externally The three vertices of a parallelogram taken in order are (–1, 0), (3, 1) and (2, 2) respectively. Find the coordinates of the fourth vertex. Ans. (–2, 1)
- 7. CENTROID If (x1, y1), (x2, y2) and (x3, y3) be the vertices of a ABC , then the co-ordinates of the centroid of ABC are : x1 x 2 x 3 y1 y2 y3 , 3 3
- 8. INCENTRE If (x1, y1), (x2, y2), (x3, y3) be the vertices A, B, C respectively of ABC with sides BC, CA and AB as a, b, c, then the co-ordinates of the incentre of ABC are : ax1 bx 2 cx 3 ay1 by 2 cy3 , a b c a b c
- 9. EXAMPLES Two vertices of a triangle are (3, –5) and (–7, 4). If the centroid is (2, –1), find the third vertex. Ans. (10, – 2) Find the coordinates of the center of the circle inscribed in a triangle whose vertices are (– 36, 7), (20, 7) and (0, – 8) Ans. (–1, 0)
- 10. SLOPE The slope (or gradient) of the line is the tangent of the angle, which the line makes with the positive direction of x-axis. The slope of the line is generally denoted by m. If A (x1, y1), B (x2, y2) are two points of a line l, not parallel to y-axis, then the slope of l, denoted by m, is the ratio = . y 2 y1 , x1 x 2 x 2 x1 Two lines (not parallel to y-axis) with slopes m1 and m2 are perpendicular iff m1m2 -1. Two lines (not parallel to y-axis) are parallel iff their slopes are equal, i.e. m1 m 2.
- 11. EXAMPLES
- 12. ANGLE BETWEEN TWO INTERSECTING LINES The positive angle between lines l1 and l2 with slopes m1 tan m2 tan respectively is given by: m1 m 2 tan 1 m1m 2
- 13. EXAMPLES
- 14. AREA OF A TRIANGLE The area of a triangle with vertices P (x1, y1), Q (x2, y2), and R (x3, y3) is given by : x1 y1 1 1 1 x2 y2 1 x1 y 2 x 2 y1 x 2 y3 x3 y2 x 3 y1 x1 y3 2 2 x3 y3 1
- 15. EXAMPLES
- 16. EXAMPLES
- 17. CHOICE OF AXES Sometimes geometrical problems can be made simple with proper choice of axes. For the proper choice of axes, we follow the rules as given below : Rule I : Whenever two perpendicular lines are given in a problem, take these lines as co-ordinate axes. Rule II : Whenever two fixed points A and B are given in a problem, we take mid-pint O of [AB] as origin and AB ad x-axis and a line through O and perpendicular to AB as y-axis.
- 18. LOCUS Definition : The locus of a point is the path traced out by the point under certain geometrical condition/conditions. Equation of locus is an equation in x and y, which is satisfied by the co-ordinates of any point on the locus and by the co-ordinates of no other point. To obtain the equation of the locus :i. Let P (x, y) by any point on the locus.ii. Apply the given condition/ conditions along with a frame, if possible.iii. Eliminate the unknown, if any, and simplify.
- 19. FORMS OF EQUATIONS OF STRAIGHT LINE Lines Parallel to Axes : Equation of straight line parallel to x-axis at a distance „a‟ is y a and equation of straight line parallel to y-axis at a distance „b‟ is x b. Point-Slope Form : The equation of the straight line passing through the point (x1, y1) and having slope m is : y - y1 = m (x - x1). Two-Point Form : The equation of the straight line passing through two point P (x1, y1) and Q (x2, y2) is : or y2 y1 y y1 x x1 y y1 x x1 x2 x1 y 2 y1 x 2 x1
- 20. Slope-Intercept Form : The equation of the straight line, which cuts off a given intercept „c‟ on y-axis and makes an angle with x-axis is : y mx c, where m tan . Intercept Form : The equation of a straight line making intercepts „a‟ and „b‟ on x-axis and y-axis respectively is . x y 1 a b General Form : The equation ax + by + c = 0 always represents a straight line, provided A and B are not both zero simultaneously.
- 21. Symmetric Form : The equation of the st. line through (x1, y1) and inclined at an angle with positive direction of x-axis in symmetrical form is : x x1 y y1 r cos sin Normal Form : The equation of the line l in terms of p, the length of perpendicular from origin on the line and angle , which this perpendicular makes with x-axis, is x cos y sin p.
- 22. EXAMPLES Find the equation of a line passing through (2, –3) and inclined at an angle of 135º with the positive direction of x-axis. Ans. x + y + 1 = 0 Find the equation of a line with slope –1 and cutting off an intercept of 4 units on negative direction of y-axis. Ans. x + y + 4 = 0 Find the equation of the line joining the points (– 1, 3) and (4, – 2) Ans. x + y – 2 = 0
- 23. EXAMPLES
- 24. DISTANCE OF A POINT FROM A LINE Length of perpendicular segment drawn from given point (x1, y1) to ax by c 0 is : ax1 by1 c 2 2 a b
- 25. EXAMPLES
- 26. POINT OF INTERSECTION The point of intersection of straight lines a1x b1y c1 0 and a2x b2y c2 0 is : b1c2 b 2 c1 c1a 2 c2 a1 , a1b 2 a 2 b1 a1b 2 a 2 b1 where a1b2 a2b1 K 0.
- 27. CHANGE OF AXES Translation of Axes : If we shift the origin without rotation of axes, we call it is “Translation of axes”. If we shift the origin to (h, k), then : x X h and y Y k or equivalently, X x h and Y y k, where (X, Y) and (x, y) are current and old co-ordinates respectively.
- 28. Rotation of Axes : If we rotate the axes through an angle „ ‟ in the anti-clockwise direction without changing the origin, we call it is “Rotation of axes”. If we rotate the axes through an angle , then : x X cos – Y sin and y X sin + Y cos or equivalently, X x cos y sin and Y y cos x sin where (X, Y) and (x, y) are current and old co-ordinates respectively.
- 29. MORE RESULTS Relative positions of points (x1, y1) and (x2, y2) w.r.t. the straight line ax by c 0: Two points (x1, y1) and (x2, y2) lie on the same or opposite sides of the straight line ax by c 0 according as ax1 by1 c and ax2 by2 c have the same sign or opposite signs.
- 30. EXAMPLES Are the points (3, – 4) and (2, 6) on the same or opposite side of the line 3x – 4y = 8 ? Ans. Opposite sides Which one of the points (1, 1), (–1, 2) and (2, 3) lies on the side of the line 4x + 3y – 5 = 0 on which the origin lies? Ans. (–1, 2)
- 31. Bisector of the angles between two lines : The equations of the bisectors of the angles between the st. lines a1x b1y c1 0 and a2x b2y c2 0, a1b2 K a2b1 are given by : a1 x b1 y c1 a 2 x b2 y c2 2 2 2 2 a 1 b1 a 2 b 2
- 32. EXAMPLES Find the equations of the bisectors of the angle between the straight lines 3x – 4y + 7 = 0 and 12 x – 5y – 8 = 0.Ans. 21 x + 27 y – 131 = 0 & 99 x – 77 y + 51 = 0 Find the equations of the bisectors of the angles between the following pairs of straight lines 3x + 4y + 13 = 0 and 12x – 5y + 32 = 0Ans. 21x – 77y – 9 = 0 and 99x + 27y + 329 = 0
- 33. Bisector of the angle containing the origin and that of not containing the origin : We write given equations as a1x b1y c1 0 and a2x b2y c2 0, c1c2 0 , then : a1 x b1 y c1 a 2 x b 2 y c2 is the bisector of the angle 2 2 a1 b1 a 2 b2 2 2 containing the origin and a1 x b1 y c1 a 2 x b2 y c2 is the bisector of the angle 2 2 2 2 a1 b1 a2 b2 not containing the origin.
- 34. EXAMPLES For the straight lines 4x + 3y – 6 = 0 and 5x + 12y + 9 = 0, find the equation of the bisector of the angle which contains the origin. Ans. 7x + 9y – 3 = 0 Find the equation of the bisector of the angle between the lines x + 2y – 11 = 0 and 3x – 6y – 5 = 0 which contains the point (1, – 3). Ans. 3x – 19 = 0
- 35. FAMILY OF LINES Family of Lines : We know that two conditions are required in order to determine a line uniquely. Lines satisfying one condition depend on single essential constant. Such a system of lines is called a one- parameter family of lines and the undetermined constant is called the parameter. Equation of Family of Lines : The equation of the family of lines through the point of intersection of two lines : a1x b1y c1 0 and a2x b2y c2 0 is a1x b1y c1 a2x b2y c2) 0
- 36. EXAMPLES
- 37. PAIR OF STRAIGHT LINES Homogeneous Equation : The equation ax2 2hx by2 0 represents a pair of straight lines passing through the origin if h2 ab . Angle Formula : The angle between the two lines given by ax2 2hxy by2 0 is : 1 2 h 2 ab tan a b
- 38. Equation of Bisectors : The equation of the bisectors of the angles between the x 2 y2 xy lines ax2 2hxy by2 0 is a b h
- 39. General Equation : The equation ax2 2hxy by2 + 2gx + 2fy +c 0 represents a pair of lines if abc 2fgh af2 bg2 ch2 0. Assuming that the equation ax2 2hxy by2 2gx 2fy c 0 represents two straight lines, then the angle between them is 1 2 h 2 ab tan a b The lines represented by ax2 2hxy by2 2gx 2fy c 0 are perpendicular iff a b 0.
- 40. Equation of Straight Lines : Joining the origin to the points of intersection ofi. two linesii. one line and a curveiii. two curves : Make one equation homogeneous with the help of the other equation.
- 41. EXAMPLES
- 42. HOMOGENIZATION
- 43. EXAMPLES Find the equation of the straight lines joining the origin to the points of intersection of the line 3x + 4y – 5 = 0 and the curve 2x2 + 3y2 = 5. Ans. x2 – y2 – 24xy = 0 Find the equation of the straight lines joining the origin to the points of intersection of the line lx + my + n = 0 and the curve y2 = 4ax. Also, find the condition of their perpendicularity. Ans. 4alx2 + 4amxy + ny2 = 0; 4al + n = 0

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