. Plane and solid analytic geometry; an elementary textbook. be chosen through which no realparabola can be drawn. PROBLEMS 1. Find the equation of a conic through the points (a) (2, 3), (0, - 3), (2, 0), (5, 5), (- 5, - 5).(6) (5, 3), (4, 4), (2, 6), (7, 1), (0, 0).(c) (2, 4), (4, 3), (6, 2), (0, - 1), (1, 0). 2. Find the equation of a parabola through the points (a) (0, 0), (8, 8), (4, 2), (- 4, 2). (b) (0,0), (1,0), (-1,1), (-1,-1).(e) (4,3), (0,-4), (6,1), (-6,2).(cf) (12,-6), (3,0), (0,2), (-3,4). 3. Determine the nature of the conies obtained in problems1 and 2. CHAPTER XIVPROBLEMS IN LO
. Plane and solid analytic geometry; an elementary textbook. be chosen through which no realparabola can be drawn. PROBLEMS 1. Find the equation of a conic through the points (a) (2, 3), (0, - 3), (2, 0), (5, 5), (- 5, - 5).(6) (5, 3), (4, 4), (2, 6), (7, 1), (0, 0).(c) (2, 4), (4, 3), (6, 2), (0, - 1), (1, 0). 2. Find the equation of a parabola through the points (a) (0, 0), (8, 8), (4, 2), (- 4, 2). (b) (0,0), (1,0), (-1,1), (-1,-1).(e) (4,3), (0,-4), (6,1), (-6,2).(cf) (12,-6), (3,0), (0,2), (-3,4). 3. Determine the nature of the conies obtained in problems1 and 2. CHAPTER XIVPROBLEMS IN LOCI 1. Find the locus of the vertex of a right angle whose sides are tangent to the ellipse — + %? — 1. a2, b The equations of any two perpendicular tangents PKand PL may be written in the form > = l1x + Vl12a2 + b2, and y = I2x + v72%2 + b2, where ?x/2 = 1. If P is their point of intersection, itscoordinates (V, y) mustsatisfy both equations. Substituting these co-ordinates, and replacingl2 by , we have r y = lxxl + ^/l2a2 + b2, ii -f*+V£+ h I 2. Fig. 92. two equations in x\ y\and the variable param-eter lv By eliminating lv we shall obtain a single equa-tion in xf and yf. Clearing of fractions, transposing, andsquaring, 185 186 ANALYTIC GEOMETRY [Ch. XIV y2 - 2 llXy + h2x2 = l^a2 + 627^2 + 2 Z^y 4- x2 = a2 + ^2Adding, (l + ^z/^ + Cl + ^^^Cl + ^^^ + Cl-h^2)^.Dividing by (1 + Zx2), y2 + a/ = a2 + 62,or .t2 + y2 = a2 + b2. The locus is the director circle, a circle having the samecentre and Va2 + b2 as radius. 2. Find the locus of the intersection of perpendiculartangents to a parabola. 3. Find the locus of the intersection of tangents to theellipse if the product of their slopes is constant. As in problem 1, the equations connecting x\ y\ Iv andl2 are (1) y = llX + ^/l2a2 + b2, (2) yf = l2x+Vl22a2+b2, (3) ya = *. But the method of elimination used in that problemwill not apply here. Transpose and square (1) and (2), y2 - 2 l^xy + lfxn = l2a2 + 52, y2 - 2 l2xy + l2x2 = l2
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