GEOMETRY. 



The problems relatiing to a ftraight line, in general, re- tangents are the fame, that is, « rr «' : introducing tliis 



quire the unknown ([uantities a and l> to be determined, fo condition in equation (4), tliat of the line required will be 



that the required line y — ax + b {\) may have the re- _,.—/ = a [x — x') (5). 



quired pofition. If the given point be taken on the given line, then y' \i 



Pkobi EM I. *■''*-' ^'^hie cf_y, wliich correiponds to x — x' ; and equation 



, . , .■ ' r 1 1. . • • . (I) becomes 



To make a ftraight Ime pals through two given points. ,. _ ^ ^.r , r 



Let the co-ordinates of one point be A.-', J, and of the other ,-'',. '^ , 



.v", 1'" ; which fignifies that when the ordmate generally ex- Subllitutmg this value ot y m equation (5), we obtain 

 preffcd by y becomes y', the ordinate x becomes x' ; and_ y = a x 4- li 



when .V = x'', y = f , «e have therefore two equations ot ^i^i^h (Irews that the required line and the given line are 



condition : identical, which is, indeed, felf-evident. 



(2) y'z= ax' + b, y" — ax" + B (3) 



from whence we obtain _^ Problem IV. 



— ^" ~ y = y ~ y . The equation of two ftraight lines being given to deter- 



" " x" — x' x' — x" ' mine the angle which they make with each other, (^g. 6. ) 



and bv fubftitutimr for a its value, The angles C A X, C B X, are given, the angle to be found 



and by lubitituting ^_ ^_^ j^^^^^ is A C B. Put CAX = a, CBX = «';ACB=V, 



b = „ _ ., — tang, a z= a, tang. 04'= a' ; then «'=:»-)- V, and tang. 



Or equation (2) may be fubtraftod from equation (l), then ^ _ _£_-^ j^ ^^^ 



y ^ yi — a (.V — x') ; and fubftituting fur a in this equation b ^ ' i -r aa' ^ 



its value, as found above, the equation of the line required ^^^ parallel, tang. V — o, and a' = a ; if they are per- 



is I 



„f ,,M pcndlciilar, tancr. V = oo , or ~- = o; tlierefore 



y-y>=y^JL{x-x') (4) , ^^"g- V 



•' •' x' — x" _ I -|- aa' =z o. 



io which equation we may obferve that when x becomes x', 

 the fecond term becomes equal to zero, and_y = y'. Problem V. 



If ji' =-y", thcnjr — y', or y = y", which denotes that the "p^ determine the equation of a ftraight hne that fliall 

 line is parallel to the axis A X. pafs through a given point, and make a given angle with a 



v' — v" , , 1 • 1- ..I. V given ilraight line. Let the equation of the <riven line 

 If V = x", a = •^— ^ - = »=. tlie angle Avhich the bne |^ _b _ ^ ^ ^;^. ^ ^ ^ ^ ^^ 



makes with the axis is in this cafe a right one. and that required - y=a-x^l' (2) _ 



a and 1/ are given ; rrom whence a and b are to be determined 

 Problem II. by the conditions of the problem. Since the required hne is 



To determine the diftance between two given points in a to pafs through a point whofe co-ordinates are .y',jv', its equa- 

 1 tion muft iubhft vs hen .v := .\-', and y =x y' ; hence 



The diftance M' M" (fg. 5.) is the hypothenufe of the y' = a' x' + l>' (3), and confequently 



right-angled triangle M' M" m" : if it be reprcfen ted by D, y —y' = a' (x - x') (4) ; 



then D = ^W'-^r + W^ = '^''(7^7^+ (■v"-*')^ ^"^f"S determined from one of the conditions, it remains 



, , : , . . . J- I „ t ' k»„„.«. to determine a from the other. Now, m being the trifrono- 



Tf the Doint M is at the oriffin, the co-ordinates .r' J become . r ,. 1 i • i .1 . r i " -.u 



i.1 inc puuiu i>j IS aL L. J, , J , TVi metric tangent ot the angle which the two hues make with 



nothing, and the pre ceding expreflion is reduced to i) — b 6 a' - a 



V y'" 4- x' = x" V 1 + V, becaufe then the equation to each other (by Problem IV.), m = — , ; hence a' = 



the ftraight line is y = ax; and as this alfo takes place ^ ^^^ i +aa 



when X — x",y = y"} it gives ji" = ax". . Subftituting this value of a' in equation {4), we 



I — ma 

 Problem III. have the equation required, 



To determine the equation of a ftraight line that firall :_" + '" r 



pafs through a given point, and be parallel to a given y-y —7^-— (■*• — •'') (5)- 



ftraight line. 



Thefe two conditions arc fufficicnt to determine the two Problem VI 



elements of pofition for the required ftraight line. .j,^ determine the condition under which three lines, drawn 



Let the equation of the given ftraight line be _y = ax from the angles of a triangle, will meet in a fingle point. 



4_ I, (i ), and tliat of the line required _v = a x + b' Let x, y' ; .r", y" ; x'", y"' be the co-ordinates of the 



(2): a and i are, in this cafe, given, and «' and ^' re- angles D, B, C {fg- I-) ; the equations to the , three 



qu'ued. Lety,j/' denote the co-ordinates of the given point ; ftraight lines drawn from tliefe angles will be j/ — ^y' :^ a', 

 andfince this point is in the required line, we have y' — (V — x);j— jv'= a- {x — x") ; y ^ f' — «"■ (.v-.v'"); 

 ^' x' 4 b (3)- a, a', n'" being the trigonometrical tangents of the angles 



From equation (l) take equation (3), and y — y' = a' which they make with the axis AX. "For thefe lines to 

 Ix — .■< ) (4)' meet in a point, it is reipiifite that the fame fyftem of values 



The fame refult may be obtained by taking the value oib' of x,y, ftiould ftibfift for tlie three equations, which is cquiva- 

 in equation (3), and fubftituting it in equation (2). More- lent to finding the value of ,r and _v by means of any two of 

 over the two urai^ht lines being parallel, their trigonometric them, and making the refult of their fublUtiitiun in the 



third 



