EXERCISES. Ex. 866. If the abscissa of a point is equal to its ordinate, each is equal to the latus rectum. Ex. 867. If a secant PP meets the directrix at H, then HF is the bisector of the exterior angle between the focal radii FP and FP'. A straight line that cuts the curve is called a secant. Ex. 868. To draw a tangent and a normal at a given point of a parabola. Ex. 872. The tangent at any point meets the directrix and the latus rectum produced at points equally distant from the focus. Ex. 873. The circle whose diameter is FP touches the tangent at A. Ex. 874. The directrix touches the circle that has any focal chord for diameter. Ex. 875. Given two points and the directrix, to find the focus. Ex. 876. The perpendicular FC bisects TP. (See figure, page 414.) Ex. 877. Given the focus and the axis, to describe a parabola which shall touch a given straight line. Ex. 878. If PN is any normal, and the triangle PNF is equilateral, then PF is equal to the latus rectum. Ex. 879. Given a parabola, to find the directrix, axis, and focus. Ex. 880. To find the locus of the centre of a circle which passes through a given point and touches a given straight line. Ex. 881. Given the axis, a tangent, and the point of contact, to find the focus and directrix. Ex. 882. Given two points and the focus, to find the directrix. Ex. 883. The triangles formed by the two tangents from any point and the focal radii to the points of contact are similar. Ex. 884. If a diameter of a parabola is cut by a chord and the tangent at either end of the chord, the segments of the diameter between the tangent and the chord made by the curve are in the same ratio as the segments of the chord. THE ELLIPSE. 896. DEF. An ellipse is a curve which is the locus of a point that moves in a plane so that the sum of its distances from two fixed points in the plane is constant. 897. DEF. The fixed points are called the foci, and the straight lines which join a point of the curve to the foci are called the focal radii of that point. 898. The constant sum of the focal radii is denoted by 2 a, and the distance between the foci by 2 c. 899. DEF. The ratio c:a is called the eccentricity, and is denoted by e. Therefore, c = ae. 900. COR. 2 a must be greater than 2c (§ 138); hence, e must be less than 1. 901. The curve may be described by the continuous motion of a point, as follows: Fasten the ends of a string whose length is 2 a at the foci F and F. Trace a curve with the point P of a pencil pressed. against the string so as to keep it stretched. The curve thus traced will be an ellipse whose foci are F and F', and the constant sum of whose focal radii is FP + PF'. The curve is a closed curve extending around both foci; if A and A' are the points in which the curve cuts FF" produced, then AA' equals the length of the string. PROPOSITION XI. PROBLEM. 902. To construct an ellipse by points, having given the foci and the constant sum 2 a. Let F and F be the foci, and CD equal a. Through the foci F, F" draw a straight line; bisect FF" at 0. Lay off OA' equal to OA equal to CD. Then A and A' are two points of the curve. and 2 a, A'F + A'F' = A'F + AF = AA' = 2 d. To locate other points, mark any point X between F and F'. Describe arcs with F as centre and AX as radius; also other arcs with F" as centre and A'X as radius; let these arcs cut in P and Q. Then P and Q are two points of the curve. This follows at once from the construction and § 896. By describing the same arcs with the foci interchanged, two more points R, S may be found. By assuming other points between F and F', and proceeding in the same way, any number of points may be found. The curve passing through all the points is an ellipse having F and F for foci, and 2 a for the constant sum of focal radii. Q. E. F. 903. COR. 1. By describing arcs from the foci with the same radius OA, we obtain two points B, B' of the curve which are equidistant from the foci. Therefore the line BB' is pendicular to AA' and passes through O. 904. DEF. The point O is called the centre. per § 161 The line AA' is called the major axis; its ends A, A' are called the vertices of the curve. The line BB' is called the minor axis. The length of the minor axis is denoted by 26. 905. COR. 2. The major axis is bisected at 0, and is equal to the constant sum 2 a. 906. COR. 3. The minor axis is also bisected at 0. § 161 907. COR. 4. The values of a, b, c are so related that For, in the rt. ▲ BOF, a2 = b2 + c2. BF2 = OB2 + OF2. § 371 908..COR. 5. The ellipse is symmetrical with respect to its major axis. For the axis AA' bisects PQ at right angles. § 161 909. DEF. The distance of a point of the curve from the minor axis is called the abscissa of the point, and its distance from the major axis is called the ordinate of the point. 910. DEF. The double ordinate through the focus is called the latus rectum or parameter. NOTE. In the following propositions F and F' denote the foci of the ellipse, O the centre, AA' the major axis, and BB' the minor axis. PROPOSITION XII. THEOREM. 911. An ellipse is symmetrical with respect to its minor axis. Let P be a point of the curve, PDQ be perpendicular to OB, meeting OB in D, and let DQ equal DP. To prove that Q is also a point of the curve. Proof. Join P and Q to the foci F, F'. Revolve ODQF about OD; F will fall on F', and Q on P. 912. DEF. Every chord that passes through the centre of an ellipse is called a diameter. 913. COR. 1. From §§ 908, 911 it follows that an ellipse consists of four equal quadrantal arcs symmetrically placed about the centre. § 213 914. COR. 2. Every diameter is bisected at the centre. § 209 |