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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' — —

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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.

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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.

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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

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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.

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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

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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.

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213. We may now ascertain the form of the hyperbola. Take the equation referred to the centre as origin,

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For every value of x less than a, y is impossible. When For every value of x greater than a there

x = a, y = 0.

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