Definition. An Ellipse is the locus of a point which moves so that its distance from a fixed point, called the focus, bears a constant ratio, which is less than unity, to its distance from a fixed line, called the directrix. 108. To find the equation of an ellipse. Let S be the focus and KL the directrix. suppose. =e:1 There will be a point A' in ZS produced such that Let C be the middle point of A A', and let AA'= 2a. Then SA-e.AZ, and SA' = e. A'Z; .. SA + SA' = e (AZ+A'Z) ; .. 2AC=2e. ZC; . (ii). Now let C be taken as origin, CA' as the axis of x, and a line perpendicular to CA as the axis of y. Let P be any point on the curve, and let its coordinates be x, Y. Then, in the figure, Putting =0, we get 1 (iii). y=±a √√(1 − e2); which gives us the intercepts on the axis of y. If these lengths be called ±b, we have ..........(iv), 1 .(v). 8 The latus rectum is the chord through the focus parallel to the directrix. To find its length we must put x = — ae in equation (v). so that the length of the semi-latus rectum is a 109. In equation (v) [Art. 108] the value of y cannot be greater than b, for otherwise would be negative; and similarly x cannot be greater than a. Hence an ellipse is a curve which is limited in all directions. If x be numerically less than a, y will be positive; and for any particular value of x there will be two equal and opposite values of y. The axis of a therefore divides the curve into two similar and equal parts. So also, if y be numerically less than b, x2 will be positive, and for any particular value of y there will be two values of x which will be equal and opposite. The axis of y therefore divides the curve into two similar and equal parts. From this it follows that if on the axis of a the points S', Z' be taken such that CS' SC, and CZ' = ZC, the point S' will also be a focus of the curve, and the line through Z' perpendicular to CZ' will be the corresponding directrix. If (x, y) be any point on the curve, the co-ordinates x2, y2 x', y will satisfy the equation + -1=0; and it is a2 b2 clear that in that case the co-ordinates — x', — y' will also satisfy the equation, so that the point (-x', -y) will also be on the curve. But the points (x, y) and (-x', — y') are on a straight line through the origin and are equidistant from the origin. Hence the origin bisects every chord which passes through it, and is therefore called the centre of the curve. The chord through the foci is called the major axis, and the chord through the centre perpendicular to this the minor axis. 110. To find the focal distances of any point on an ellipse. also In the figure to Art. 108, since SP = ePM, we have An ellipse is sometimes defined as the locus of a point which moves so that the sum of its distances from two fixed points is constant. To find the equation of the curve from this definition. Let the constant sum be 2a, and the distance between the two fixed points be 2ae. Take the middle point of the line joining the fixed points for origin, and this line and a line perpendicular to it for axes, then we have from the given condition √(x -ae)2+ y2+ √(x+ae)2+ y2=2a which, when rationalized, becomes y2+x2 (1-e2)=a2 (1 − e2), which is the equation previously obtained. 111. The polar equation of the ellipse referred to the centre as pole will be found by writing r cose for x, and r sine for y in the equation The equation (i) can be written in the form is positive, we see from (ii) that the least 112. We have found that for all points on the ellipse We can shew in a manner similar to that adopted in Art. 92 that, if x, y be the co-ordinates of any point within x2 y2 a2 b2 – 1 will be negative, and that + 1 x2 y2 the curve, + a2 b2 will be positive if x, y be the co-ordinates of any point outside the curve. 113. To find the points of intersection of a given straight line and an ellipse, and to find the condition that a given straight line may touch the ellipse. NOTE. We shall henceforth always take x2 y2 + = 1 as a2 bu the equation of the ellipse, unless it is otherwise expressed. Let the equation of the straight line be At points which are common to the straight line and the ellipse both these relations are satisfied. the common points we have or Hence at ...... x2 (b2 + a3m2) + 2mca3x + a2 (c2 — b2) = 0 ................ (ii). This is a quadratic equation, and every quadratic equation has two roots, real, coincident, or imaginary. Hence there are two values of x, and the two corresponding values of y are given by equation (i). |