. Algebraic geometry; a new treatise on analytical conic sections . en C = 0, (iv) when A = 00, (v) when A = C=0, (vi) when B = a>. 26. Find the equation of the straight line through the origin and themiddle point of the line joining (2, 3), (4, - 9). 27. Find the equation of the straight line through the origin and themiddle point of the line joining (x-^, y{) and {x^, 2/2)- 28. What is the value of m if the straight line y=ma; 4-4 passes throughthe point of intersection oi x-y = 5 and x + y=7. 29. Prove that the following three straight lines are concurrent: x-y=6, 4y-3a: + 22=0, 6a; + 5y
. Algebraic geometry; a new treatise on analytical conic sections . en C = 0, (iv) when A = 00, (v) when A = C=0, (vi) when B = a>. 26. Find the equation of the straight line through the origin and themiddle point of the line joining (2, 3), (4, - 9). 27. Find the equation of the straight line through the origin and themiddle point of the line joining (x-^, y{) and {x^, 2/2)- 28. What is the value of m if the straight line y=ma; 4-4 passes throughthe point of intersection oi x-y = 5 and x + y=7. 29. Prove that the following three straight lines are concurrent: x-y=6, 4y-3a: + 22=0, 6a; + 5y + 8 = 0. 26. Given a point (jBj, yj) on a straight line, and the slope (m) ofthe line, to find its equation. Let PQ be the line, Q being the given point (x^, y-^. Also letthe line make an angle tan~i??i with Ox. If P(a;, y) is any point on the line, it is required to find theequation connecting x, y, Xj, y-^, and m. 28 THE STRAIGHT LINE. [chap. II. Draw the ordinates PN, QM ; and draw QK perpendicular to PN - KN = PN - QIVl =y-j/j,QK= MN = ON - OM =a;-Ki;. from the A PKQ, —- = tan PQK = m,QK V-V^or i_-ii = j7ij ■■ y - yi = ni(x - Xj) is the reqd. equation. 27. The following analytical method might be used. Let y — mx + chB the equation required, where c is hypothesis, the point (Xj, y^ lies on the line. .. »!, ^1 satisfy the equation y = mx + c;.. yi = mx^ + y = mx + c; :. by subtraction, y -yi = m(x-x^) is the reqd. equation. 28. To find the eqmtwn of the straight line passing through thetwo given points (x^, «/j), {x^, y^). Let A and B be the given points, (x^, y^), (x^, y^), and P{x, y) anypoint on the line, It is required to find the equation connecting x, y, x-^, y,, x,andyj. ^ ART. 29.] THE STRAIGHT LINE. 29 Draw the ordinates AH, BL, PN, and draw AKM perpendicularto BL and PN. From the similar triangles PMA, BKA, PM AM . - But BK AK PM = PN - MN = PN - AH = y - yj,BK = BL-KL=BL-AH=2/2-yj,AM = HN = ON - OH =0!-ail,AK = HL = OL-OH =
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