By Alfred Clement Jones
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Extra resources for An Introduction to Algebraical Geometry
Oblique Coordinates. (a) Let the equation of the straight line be given in the form =p XCOSOL + ycosfi and let P (x^ the point be Through P draw (i) y^). a -f y cos /3 == p. draw OLM per- a straight line parallel to x cos And from Draw pendicular to these lines. / cos fi =p+# and . hence this, therefore chapter OM = p + q of PM is hence the equation *\~ q. 3 of this by j), ~7 it t cos ! cos ft = jp + ^ a + ^ cos /i the equation of a straight line x cos a + # cos j> /:* ; p. given in the form is = 0, the length of the perpendicular from any point on it is found by It substituting its coordinates in the left-hand side of the equation.
Is (4 = 1- 2 therefore - 5x' 4- 20). 7# on 24y (7*'-24*/' -f 72 = is 72). e. (#', y') is a %, and 2#-f 11 Examples II f. 32, 1/ = 33, 2$). In the above example work out similarly the coordinates of each of the ex-centres, giving a reason for the signs chosen in each case. 2. Find the centre of the circle inscribed in the triangle whose sides are x-y + l = 3. r-3y + 5 = 0. = 12. 0. Relations between two straight lines whose equations are given. 8. To find (i) = of the sides of a triangle are x 0, 3j?
40 and from 16. D is a lines x-y A, B, C\ straight the parallelogram. 3. B the distance between A( + 4f/= 2. b y equal to the length of the required perpendicular, the point Ax -f By + C = on ~ parallelogram 7 and 3:r + A ; is the point ( -5, 2) J3C, CD are the Find the lengths of the sides of ; == 28. 4// Through a point P(h, k) a straight line is drawn, making an angle fr + fy/ = 1 at Q. r, to meet the straight line 6 with the axis of Find the coordinates of a point on Hence PQ find the locus of such points distant when PQ/n from P.