7. To express the area of a quadrilateral in terms of the co-ordinates of its angular points. Let the angular points A, B, C, D, taken in order, be (x1, Y1), (X2, Y2), (x, y) and (x, y). 3 Draw AK, BL, CM, DN parallel to the axis of y, as in the figure. Then the area ABCD - KABL+LBCM - MCDN – NDAK. = And, as in the preceding Article, 2 KABL= {(y1+y2) (x − x12), 3 Hence ABCD = {(Y1 + Y2) (x, − ∞ ̧) + (Y2+Y3) (X3 − X2) + (Y3+Y1) (x ̧ − x ̧) + (Y1+Y1) (x ̧ −x ̧)}; 4 The area of any polygon may be found in a similar manner. Ex. 1. Find the area of the triangle whose angular points are (2, 1), (4, 3) and (2,5). Also find the area of the triangle whose angular points are (4, 5), (5, 6) and (3, 1). Ans. 4, . Ex. 2. Find the area of the quadrilateral whose angular points are (1, 2), (3, 4), (5, 3) and (6, 2). Also of the quadrilateral whose angular points are (2, 2), (−2, 3), (−3, −3) and (1, −2). Ans. 11, 20. 8. If a curve be defined geometrically by a property common to all points of it, there will be some algebraical relation which is satisfied by the co-ordinates of all points of the curve, and by the co-ordinates of no other points. This algebraical relation is called the equation of the curve. Conversely all points whose co-ordinates satisfy a given algebraical equation lie on a curve which is called the locus of that equation. For example, if a straight line be drawn parallel to the axis OY and at a distance a from it, the abscissae of points on this line are all equal to the constant quantity a, and the abscissa of no other point is equal to a. Hence x = a is the equation of the line. Conversely the line drawn parallel to the axis of y and at a distance a from it is the locus of the equation x = α. Again, if x, y be the co-ordinates of any point P on a circle whose centre is the origin 0 and whose radius is equal to c, the square of the distance OP will be equal to x+y2 [Art. 4]. But OP is equal to the radius of the circle. Therefore the co-ordinates x, y of any point on the circle satisfy the relation 2+y=c. That is, x2 + y2= c2 is the equation of the circle. Conversely the locus of the equation x2 + y2 = c2 is a circle whose centre is the origin and whose radius is equal to c. In Analytical Geometry we have to find the equation which is satisfied by the co-ordinates of all the points on a curve which has been defined by some geometrical property; and we have also to find the position and deduce the geometrical properties of a curve from the equation which is satisfied by the co-ordinates of all the points on it. An equation is said to be of the nth degree when, after it has been so reduced that the indices of the vari CO-ORDINATES. 9 ables are the smallest possible integers, the term or terms of highest dimensions is of n dimensions. For example, the equations axy + bx + c = 0, x2+xy √ a+b3 = 0, and √x+y=1 are all of the second degree. Ex. 1. A point moves so that its distances from the two points, (3, 4), and (5, 2) are equal to one another; find the equation of its locus. Ans. x-3y=1. Ex. 2. A point moves so that the sum of the squares of its distances from the two fixed points (a, 0) and (−a, 0) is constant (2c2); find the equation of its locus. Ans. x2+y2=c2 — a2. Ex. 3. A point moves so that the difference of the squares of its distances from the two fixed points (a, 0) and a, 0) is constant (c2); find the equation of its locus. Ans. 4ax ±c2. ¿Ex. 4. A point moves so that the ratio of its distances from two fixed points is constant; find the equation of its locus. Ex. 5. A point moves so that its distance from the axis of x is half its distance from the origin; find the equation of its locus. Ans. 3y2-x2=0. 9. The position of a point on a plane can be defined by other methods besides the one described in Art. 1. A useful method is the following. If an origin O be taken, and a fixed line OX be drawn through it; the position of any point P will be known, if the angle XOP and the distance OP be given. r X These are called the polar co-ordinates of the point P. usually denoted by r, and the angle XOP is called the vectorial angle, and is denoted by 0. The angle is considered to be positive if measured from OX contrary to the direction in which the hands of a watch revolve. The radius vector is considered positive if measured from O along the line bounding the vectorial angle, and negative if measured in the opposite direction. If PO be produced to P', so that OP' is equal to OP in magnitude, and if the co-ordinates of P be r, 0, those of P' will be either r, π + 0 or - r, 0. 10. To find the distance between two points whose polar co-ordinates are given. Let the co-ordinates of the two points P, Q be and k2, 02. Then, by Trigonometry, PQ2 = OP2 + O Q2 - 2OP. OQ cos POQ. Ꮎ ; 19 1 But OP =r1, OQ = r ̧ and < POQ= ≤ XOQ— ≤ XOP=02—01; 2 The polar equation of a circle whose centre is at the point (a, a) and whose radius is c, is c2=a2+r2 - 2ar cos (0 - a); where r, are the polar co-ordinates of any point on it. 11. To find the area of a triangle having given the polar co-ordinates of its angular points. R X Let P be (r,, 0), Q be (r,, 0), and R be (r,, 0). Then area of triangle PQR = ▲ POQ+A QOR− ▲ POR, 1 .:. ▲ PQR = {r, r, sin (0, − 01) +, r, sin (0, — 02) 2 2 +r, r, sin (0,- 0 ̧)}· 12. To change from rectangular to polar co-ordinates. Y N X If through O a line be drawn perpendicular to OX, and OX, OY be taken for axes of rectangular co-ordinates, we have at once and x= ON= OP cos XOP=r cos 0, y = NP = OP sin XOP=r sin 0. 'Ex. 1. What are the rectangular co-ordinates of the points whose polar co-ordinates are π (1, 5), (2,) and (-4, -1) respectively? 2 n Ex. 2. What are the polar co-ordinates of the points whose rectangular co-ordinates are (−1, − 1), (− 1, 、/3) and (3, −4) respectively? Ex. 3. Find the distance between the points whose polar co-ordinates are (2, 40o) and (4, 100o) respectively. 253 Ex. 4. Find the area of the triangle the polar co-ordinates of whose angular points are (1, 0), (1, 1) and (√2, 7) respectively. 2 |