Reprinted with corrections Crown 8vo. IN the following work I have tried to present the elements of Coordinate Geometry in a manner suitable for Beginners and Junior Students. The present book only dealsi with Cartesian and Polar Coordinates. Within these limits I venture to hope that the book is fairly complete, and that no propositions of very great importance have been omitted.

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Reprinted with corrections Crown 8vo. IN the following work I have tried to present the elements of Coordinate Geometry in a manner suitable for Beginners and Junior Students. The present book only dealsi with Cartesian and Polar Coordinates. Within these limits I venture to hope that the book is fairly complete, and that no propositions of very great importance have been omitted.

The Straight Line and Circle have been treated more fully than the other portions of the subject, since it is generally in the elementary conceptions that beginners find great difficulties. There are a large number of Examples, over in all, and they are, in general, of an elementary character. The examples are especially numerous in the earlier parts of the book. I am much indebted to several friends for reading portions of the proof sheets, but especially to Mr W.

Dobbs, M. For any criticisms, suggestions, or corrections, I shall be grateful. July 4, For the Second Edition the time at my disposal has only allowed me to correct the misprints that have been kindly pointed out to me by many correspondents. June 30, Polar Coordinates. Equations involving an arbitrary constant Examples of loci.. Angle between two lines given by one equation General equation of the second degree VII.

The Circle Equation to a tangent. Pole and polar.. Equation to a circle in polar coordina Equation referred to oblique axes Equations in terms of one variable IX. Orthogonal circles. Radical axis Coaxal circles X. Conic Sections. Some properties of the parabola Pole and polar Diameters. Equations in terms of one variable PAGE Auxiliary circle and eccentric angle Equation to a tangent Some properties of the ellipse Pole and polar Conjugate diameters.. Four normals through any point Examples of loci.

Quadratic Equations. If any equation be written so that the coefficient of the highest term is unity, it is shewn in any treatise on Algebra that 1 the sum of the roots is equal to the coefficient of the second term with its sign changed, 2 the sum of the products of the roots, taken two at a time, is equal to the coefficient of the third term, 3 the sum of their products, taken three at a time, is equal to the coefficient of the fourth term with its sign changed, and so on.

The quantity bb2 is called a deterniinant of the second order and stands for the quantity ab62 - a2b1, so that a1, a2 b1, 6,:: za a2b,. The quantity b6, b2, Thus, if in 1 we omit the row and column to which a, belongs, we have left the determinant b2 3 and this is the C2, C I coefficient of al in 2. The rule for finding the value of a determinant of the fourth order in terms of determinants of the third order is clearly the same as that for one of the third order given in Art.

Similarly for determinants of higher orders. A determinant of the second order has two terms. One of the third order has 3 x 2, i. One of the fourth order has 4 x 3 x 2, i. There must be some relation holding between the four coefficients a, a2, bi, and b2. The result 3 is the condition that both the equations 1 and 2 should be true for the same values of x and y..

The process of finding this condition is called the eliminating of x and y from the equations I and 2 , and the result 3 is often called the eliminant of 1 and 2. Using the notation of Art. By dividing each equation by z we have three equations between the two unknown quantities and. Two of X z these will be sufficient to determine these quantities. By substituting their values in the third equation we shall obtain a relation between the nine coefficients.

Or we may proceed thus. This is the result of eliminating x, y, and z from the equations 1 , 2 , and 3. But, by Art. Ci C2 C3 This eliminant may be written down as in the last article, viz. If again we have the four equations a,.

It will be noted that the right-hand member of each of the above equations is zero. Let OX and OY be two fixed straight lines in the plane of the paper. The line OX is called the axis of x, the line OY the axis of y, whilst the two together are called the axes of coordinates. The point 0 is called the origin of coordinates or, more shortly, the origin. Distances measured parallel to OX are called x, with or without a suffix, e.

If the distances OM and MP be respectively x and y, the coordinates of P are, for brevity, denoted by the symbol X, y Conversely, when we are given that the coordinates of a point P are x, y we know its position. In Analytical Geometry we have the same rule as to signs that the student has already met with in Trigonometry.

Finally, if P4 lie in the fourth quadrant its abscissa is positive and its ordinate is negative. Lay down on paper theposition of the points i 2, -1 , ii -3, 2 , and iii -2, These three points are respectively the points P4, P2, and P3 in the figure of Art. When the axes of coordinates are as in the figure of Art.

In general, it is however found to be more convenient to take the axes OX and OY at right angles. They are then said to be Rectangular Axes. It may always be assumed throughout this book that the axes are rectangular unless it is otherwise stated.

The system of coordinates spoken of in the last few articles is known as the Cartesian System of Coordinates. It is so called because this system was first introduced by the philosopher Des Cartes. There are other systems of coordinates in use, but the Cartesian system is by far the most important.

To find the distance between two points whose co ordinates are given. Let P1 and P2 be the two given points, and let their coordinates be respectively x,, yi and x, Y2. We therefore have [Trigonometry, Art. This follows from 2 by making both x2 and Y2 equal to zero. The formula of the previous article has been proved for the case when the coordinates of both the points are all positive.

Similarly any other case could be considered. To find the coordinates of the point which divides in a given ratio mIz the line joining two given points x1, y and X2, Y2.

This is the well-known theorem of Ptolemy. Hence prove that the medians of a triangle meet in a point. Let the coordinates of the vertices A, B, and C be x1, y, , x2, y2 , and x3, y, respectively. By the last article 2x. In the same manner we could shew that these are the coordinates of the points that divide BE and CF in the ratio 2: 1. This point is called the Centroid of the triangle. Find the distances between the following pairs of points.

Lay down in a figure the positions of the points 1, -3 and - 2, 1 , and prove that the distance between them is 5. Find the value of xi if the distance between the points. A line is of length 10 and one end is at the point 2, - 3 ; if the abscissa of the other end be 10, prove that its ordinate must be 3 or Prove that the points - 2, - 1 , 1, 0 , 4, 3 , and 1, 2 are at the vertices of a parallelogram.

Prove that the points 2, -2 , 8, 4 , 5, 7 , and -1, 1 are at the angular points of a rectangle. Prove that the point - -, 9 is the centre of the circle circumscribing the triangle whose angular points are 1, 1 , 2, 3 , and - 2, 2.

Find the coordinates of the point which The line joining the points 1, - 2 and - 3, 4 is trisected; find the coordinates of the points of trisection. The line joining the points - 6, 8 and 8, -6 is divided into four equal parts; find the coordinates of the points of section.

Find the coordinates of the latter point of section. Prove that the lines joining the middle points of opposite sides of a quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one another. A, B, C, D Prove that a point can be found which is at the same distance from each of the four points amn, , am2,, am3,, da and amnm2mn3 To prove that the area of a trapezium, i.

AL -I.. To find the area of the triangle, the coordinates of whose angular points are given, the axes being rectangnlar.

If we use the determinant notation this may be written as in Art. In order that the expression for the area in Art.

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