1.

EXAMPLES ON CHAPTER III.

Draw a parabola, the focus F, the position of the axis (FT) and a tangent (PT) being given.

(From F draw FY perpendicular to PT meeting it in Y, and from Y draw YA perpendicular to FT meeting it in A. A will be the vertex of the required parabola.)

2. Draw a parabola, the focus F, a tangent PT and the length of the latus rectum being given.

(With centre F and radius equal to one-fourth of the given latus rectum, describe a circle; from F draw FY perpendicular to the given tangent meeting it in Y, and from Y draw tangents to the circle. Either point of contact will be the vertex of the required parabola (two solutions). The given tangent must not cut the circle.)

3.

Draw a parabola, two points (P, Q), the tangent at one of them (PT), and the direction of the axis being given.

(Bisect PQ in V, draw VT parallel to given direction of axis meeting the given tangent in T; QT is the tangent at Q, and problem reduces to Prob. 49.)

4. Draw a parabola, the vertex (P) of a diameter, and a corresponding double ordinate QQ, being given.

1

(Bisect QQ, in V. PV will be a diameter; on VP produced make PT=PV. TQ and TQ, are the tangents at Q and Q,, and problem reduces to Prob. 49.)

1

5. Draw the locus of the foci of the parabolas which have a common vertex (A) and a common tangent PT.

(The parabola which has A for vertex, the perpendicular on PT as axis, and the distance of PT from A as latus rectum.)

6. Inscribe in a given parabola a triangle having its sides. parallel to three given straight lines AB, BC, CA.

(Draw BD parallel to the axis of the parabola meeting AC in

D and CE parallel to the axis meeting AB in E. Draw a tangent to the parabola parallel to DE (p. 61) and from P its point of contact draw PQ, PR parallel to AB, AC meeting the parabola again in Q, R. PQR will be the required triangle.)

7. Draw a parabola with a given focus, and to touch a given circle at a given point.

[Let F be the focus, P the point on the circle, draw PT the tangent, and construct an angle TPM the angle FPT. The axis of the required parabola will be parallel to PM.]

8. Shew that if tangents be drawn to a parabola from any point O, and a circle be described with the focus as centre, passing through O and cutting the tangents in P and Q, PQ will be perpendicular to the axis, and its distance from O is twice its distance from the vertex.

9. Draw a circle to touch a parabola in P, and to pass through the focus. Let it meet the parabola again in Q and Q,: draw a focal chord parallel to the tangent at P, and shew that the circle on this chord as diameter will pass through Q, Q1, and that the focal chord and QQ, will intersect on the directrix.

1

10. Draw any right-angled triangle DEF (E being the right angle). Describe a parabola with focus F and to touch ED at D, and shew that if any circle be described to pass through D and F and cutting ED produced in P, the tangent to it at P will also be a tangent to the parabola.

11. Given two lines PR, QR, and a point P on one of them, shew that any point on the circumference of the circle passing through P and R and touching QR may be taken as the focus of a parabola passing through P and to which the given lines shall be tangents.

12. AB is the diameter of a circle; with A as focus and any point on the semi-circumference of which A is the centre as foot of directrix describe a parabola, and shew that it will touch the diameter perpendicular to AB.

13. If APC be a sector of a circle of which the radius CA is fixed, and a circle be described touching the radii CA, CP and the arc AP, shew that the locus of the centre of this circle is a parabola and describe it.

14. Given a segment of a circle, describe the parabola which is the locus of the centres of the circles inscribed in it.

15. If from a point P of a circle PC be drawn to the centre, and R be the middle point of the chord PQ drawn parallel to a fixed diameter ACB, describe the locus of the intersection of CP, AR, and shew that it is a parabola.

16. Describe a parabola with latus rectum = 2·7 units, and in it draw a series of parallel chords inclined at 60° to the axis. Shew that the locus of the point which divides each chord into segments containing a constant rectangle = 4 sq. units in area, is a parabola, the axis of which coincides with the axis of the original parabola and with the latus rectum = 2·1 units.

17. Draw a parabola to touch the three sides of a given triangle, one of them at its middle point; and shew that the perpendiculars drawn from the angles of the triangle upon any tangent to the parabola are in harmonical progression.

18. Given two unequal circles (centres G and g, radii R and r) touching each other externally, from G the centre of the larger circle make GN on Gg towards g

=

R-r 2

Draw NP perpen

dicular to Gg meeting the circle in P and describe a parabola with Gg as axis and to touch the circle in P (Prob. 47), and shew that it will also touch the smaller circle.

19. Given a point F and two straight lines intersecting in 0; describe a parabola with F as focus and to touch the given lines (Prob. 42); and shew that if any circle be described passing through O and F and meeting the lines in P and Q, PQ will be a tangent to the parabola.

20.

Draw the parabola which is the locus of the centre of a circle passing through a given point and cutting off a constant intercept on a given straight line.

(The point is the focus and a perpendicular to the line the axis.)

21. Given four tangents to a parabola, shew that the directrix is the radical axis of the system of circles described on the diagonals of the quadrilateral as diameters.

22. Given the focus F, a point P on the curve and a point L on the directrix, describe the parabola.

[Tangents from L to the circle described with centre P and radius PF are the directrices of two parabolas fulfilling required conditions.]

23. Given a focus F, a tangent PT, and a point L on the directrix, describe the parabola.

[From F draw a perpendicular FY to PT meeting it in Y; produce FY to ƒ and make Yƒ = FY : ƒ is a second point on the directrix.]

24. Given three tangents to a parabola and a point on the directrix, draw the curve.

[The ortho-centre of the triangle formed by the tangents is a second point on the directrix.]

CHAPTER IV.

THE ELLIPSE.

THE ellipse has already been defined (p. 56) as the locus of a point which moves in a plane so that its distance from a fixed point in the plane is always in a constant ratio, less than unity, to its distance from a fixed line in the plane. The corresponding definition in the case of the parabola furnishes immediately the best condition for the geometrical construction of that curve, but this is not so with the ellipse. The ellipse can be more easily constructed geometrically from a property which will be shewn immediately to be involved in the above definition, and in virtue of which the curve may be defined as follows:

DEF. The ellipse is the locus of a fixed point on a line of constant length moving so that its extremities are always on two fixed straight lines perpendicular to each other.

19

In Fig. 55 let ACA1, BCB, be two straight lines intersecting each other at right angles in C. If a length (as ab) be marked off on the smooth edge of a slip of paper, and the slip be moved round so that the point a is always on the line BCB, and the point b on ACA1, then any point as P on the edge of the paper will trace out an ellipse. When the edge of the slip coincides with ACA, the tracing point will evidently be at a distance CA from C equal to aP, and when it coincides with BCB, the tracing point will be at a distance CB from C equal to bP. By this method of construction the curve is evidently symmetrical about both the lines ACA, and BCB1, i.e. if CA1 be made equal to CA, A, will be a point on the curve, and if CB1 be made equal

1

1