. Algebraic geometry; a new treatise on analytical conic sections . roots of this quadratic, m^m^ = -\; .. r= -1, or h + a = 0. But (Ji, k) is any point,on the locus. .. x + a = Q is the equation of the locus, and represents thedirectrix. Again, we may take (- a, k) as the co-ordinates of any pointon the locus, and the equation of the polar of this point is ky = 2a(a; - a). \j)y-^ = 2a{x + aij)] This straight line passes through the focus (a, 0). 142 LOCUg PBOBLEMS ON THE PARABOLA, [chap. viii. Corollary. If f-^, —\ are the co-ordinates of one end of a\m^ m)focal chord, m is the slope o
. Algebraic geometry; a new treatise on analytical conic sections . roots of this quadratic, m^m^ = -\; .. r= -1, or h + a = 0. But (Ji, k) is any point,on the locus. .. x + a = Q is the equation of the locus, and represents thedirectrix. Again, we may take (- a, k) as the co-ordinates of any pointon the locus, and the equation of the polar of this point is ky = 2a(a; - a). \j)y-^ = 2a{x + aij)] This straight line passes through the focus (a, 0). 142 LOCUg PBOBLEMS ON THE PARABOLA, [chap. viii. Corollary. If f-^, —\ are the co-ordinates of one end of a\m^ m)focal chord, m is the slope of the tangent at this point; .. (m^, - 1am) are the co-ordinates of the other end, forthe tangents at the ends are at right angles. 155. Find, the locus of the middle points of the focal radii of theparabola y^ = 4aa;. Let P be any point on the curve, and let (—^, —j be itsco-ordinates. Join SP, and bisect it at Q. We have to find the locus of (x, y) be its co-ordinates of S are (a, 0), those of P(^,^«); .. the co-ordinates of Q are. ■^^=10+^) (^> and y = — .(2) Via. 95. To find the locus of Q we haveto eliminate m between theseequations. From (2) i = ?^;ma .. substituting in (1) 2x = a(l+^\ or y^ = 2a(x-^\ is the equation of the locus. This is a parabola, whose vertex is at the point (-, 0\ whoselatus rectum is 2«, and it is co-axial with the given curve. Exs. viii. c] LOCUS PEOBLEMS ON THE PAEABOLA. 143 Examples VIII. c. [It is generally adyisable to use (-^, -^1 as the co-ordinates of anypoint on the parabola y^=4ax.] Vm^ ^J 1. Find the loous of the middle points of the ordinates of a the locus. 2. Find the equation of, and draw, the loous of the middle points ofchords of the paranoia y=iax through the vertex. 3. Through P, any point on a parabola, PQ is drawn parallel to theaxis, and SQ is drawn through the focus and parallel to the tangent at the locus of Q. 4. In any parabola, the perpendicular on the directrix fr
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