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.
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
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
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
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)
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
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
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)
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.
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.
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.
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
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.
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.
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
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.
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.
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.
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.
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)
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
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
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.
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
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
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
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
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.
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.
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