44. Shew that the area of the parallelogram in the preceding question = ay' + bx' — ab, where x', y', are the coordinates of P; and find the greatest value of this area. 45. If a line be drawn from the focus of an ellipse to make a given angle a with the tangent, shew that the locus of its intersection with the tangent will be a circle which touches or falls entirely without the ellipse according as cos a is less or greater than the excentricity of the ellipse. 46. In an ellipse SQ, HQ, drawn perpendicular to a pair of conjugate diameters intersect in Q; prove that the locus of Qis a concentric ellipse. 47. A line of constant length moves so that its ends always lie on two given lines; find the locus traced out by a point in the line which divides it in a given ratio. 48. What is represented by the equation x2 + y2 = c2 when the axes are oblique ? 49. Shew that when the ellipse is referred to any pair of conjugate diameters as axes, the condition that y = mx and y=m'x may represent conjugate diameters is mm' — — 50. The ellipse being referred to equal conjugate diameters, find the equation to the normal at any point. 51. From any point P perpendiculars PM, PN, are drawn on the equal conjugate diameters; shew that the normal at P bisects MN. CHAPTER XI. THE HYPERBOLA. 209. To find the equation to the hyperbola. The hyperbola is the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line, the ratio being greater than unity. Let H be the fixed point, YY' the fixed straight line. Draw HO perpendicular to YY'; take O as the origin, OH as the direction of the axis of x, OY as that of the axis of y. Let P be a point on the locus; join HP, draw PM parallel to OY and PN parallel to OX. Let OH=p, and let e be the ratio of HP to PN. Let x, y, be the co-ordinates of P. This is the equation to the hyperbola with the assumed origin and axes. 210. To find where the hyperbola meets the axis of x we put y = 0 in the equation to the hyperbola; thus Since e is greater than unity, 1-e is a negative quantity. e · 1 Ρ 1+ e Let OA'= 2, 04= 1, the former being measured to the left of A, then A' and A are points on the hyperbola. A and A' are called the vertices of the hyperbola, and C the point midway between A and A' is called the centre of the hyperbola. 211. We shall obtain a simpler form of the equation to the hyperbola by transforming the origin to A or C. this by 2a; hence the equation becomes y2 = (e2 − 1) (2ax' +x'2). We may suppress the accent, if we remember that the origin is at the vertex A, and thus write the equation y3 = (e2 − 1) (2ax + x2). II. Suppose the origin at C. (1). Since CA = a, we put x=x' - a and substitute this value in (1); thus y2 = (e2 — 1) {2a (x' − a) + (x' — a)2} = (e2 — 1) (x22 — a2). We may suppress the accent if we remember that the origin is now at the centre C, and thus write the equation In (2) suppose x = 0, then y2=-(e2 - 1) a2: this gives an impossible value to y, and thus the curve does not cut the axis of y. We shall however denote (e2-1) a2 by b2, and measure off the ordinates CB and CB' each equal to b, as we shall find these ordinates useful hereafter. |