38. If PSP be any parameter of a parabola whose focus is S and latus rectum L, prove that

4SP. Sp=L(SP+Sp).

39. Two tangents to a parabola drawn from the same point of the directrix are at right angles to each other.

40. Show that there is only one point in the parabola at which the focal distance is perpendicular to the tangent.

In this case the property (7) may be used.

41. Draw a normal at the extremity of the latus rectum of a parabola whose equation is y=4a(x-a), and find its distance from the origin of coordinates.

42. The tangent at any point of a parabola will meet the directrix and latus rectum produced, in two points equally distant from the focus.

43. If two equal parabolas have a common axis, a straight line touching the interior parabola, and bounded by the exterior, will be bisected by the point of contact.

44. Given the radius vector at any point of a parabola and the angle it makes with the curve; find the latus rectum and the position of the vertex.

45. Show that the parameter belonging to any diameter of a parabola varies inversely as the square of the sine of the angle at which the corresponding ordinates are inclined to it.

The polar equation should be assumed in this case.

46. Describe a parabola which shall touch a given circle at a given point, and have its axis coincident with a given diameter of the circle. In this case the latus rectum and the position of the vertex of the parabola must be found. The tangent to the circle at the given point is also a tangent to the parabola at that point. The extremity of the given diameter of circle may conveniently be taken for the origin of coordinates.

47. The abscissa and double ordinate of a parabola are h and k; the diameters of its circumscribed and inscribed circles are D and d; prove that D+d=h+k.

In solving this question we find that D=4a+h and d=−4a+k.

48. If PQ be the chord of a parabola, which is a normal at P, and the tangents at P and Q intersect in a point T; prove that the line PT is bisected by the directrix.

(1.) Having given the coordinates of P, find those of Q by combining the equations to the normal at P and to the parabola. (2.) Find the coordinates of T, by combining the equations to the tangents at P and Q.

49. If BC, CD be two consecutive arcs of a parabola, the sagitta of which drawn parallel to the axis are equal; prove that the chord of BCD is parallel to the tangent at C.

Def. The sagitta of an arc is a straight line drawn from the middle point of the chord to meet the arc.

(1.) It must be shown that the chord of BCD is bisected by the line drawn through C parallel to the axis.

(2.) If a, 6 be the angles at which the chords BC, CD respectively are inclined to the axis, then BC sin a=CD sin ß.

50. If AB, BC be two consecutive arcs of a parabola, and LW, GH, MV be the sagittæ, drawn parallel to the axis of the arcs AB, AC, BC respectively; prove that VLW+ √MV= ✔GH.

The method used in the solution of the preceding question may here be repeated.

51. Find the position and magnitude of the parabola whose equation is

a

+

= 1.

52. Find the area included between the parabola y2=4ax, and the straight line xy+a.

53. A triangle is formed by the meeting of three tangents to a parabola. Show that the products of the alternate segments of the tangents made by their mutual intersections are equal.

This question may be solved by the following property: if P, Q be points in the parabola at which two of the tangents are drawn, S the focus, and B the intersection of these tangents, then PB2 : QB =SP: SQ.

54. A circle described through the intersections of three tangents of a parabola will pass through the focus.

If lines be drawn from the focus to the extremities of that tangent which lies between the other two, the angle at the focus is the supplement of the angle between those two tangents. As in the last question PB: QB2=SP : SQ; also SP. SQ=SB2: both these properties are required for this problem.

55. If one side of a triangle and two others produced be tangents to a parabola, and the points of contact be joined, a triangle will be formed whose area is double of that of the exterior triangle.

56. From the vertex of a parabola a straight line is drawn, inclined at 45° to the tangent at any point; find the equation to the curve which is the locus of their intersections.

The equation to the curve is a(y + x)2=(y2+x2) (y—x).

57. In the focal distance SP take Sp equal to the ordinate PN. Find the equation to the curve traced out by the point p.

If we assume Sp=g and <NSp=0, the locus of p is expressed by 0

the equation g2a.cot.

2

58. Two straight lines which are always tangents to a given parabola, are so inclined to the axis of x that the sum of the cotangents of the angles which they make with that axis is constant; prove that the locus of their intersections is a straight line parallel to the axis.

59. If several parabolas have the same vertex and axis, the locus of the extremities of normals to them from a given point in the axis is an ellipse.

60. From any point Q in the line BQ which is perpendicular to the axis CAB of a parabola whose vertex is A, QP is drawn parallel to the axis to meet the curve in P; show that if CA be taken equal to AB the locus of the intersections of AQ and CP is a parabola.

The required locus coincides with the original parabola.

61. If PT, QT be two tangents at the points P, Q of a parabola whose focus is S, then SP. SQ=ST; and if SP, SQ include a given angle a, the locus of T will be a hyperbola whose eccentri

62. The locus of the intersection of two normals to a parabola at

right angles to one another, is a parabola whose latus rectum is onefourth of the latus rectum of the original.

3. SP=a+ex, HP=a-ex, S, H being the foci.

4. a2yy'+b2xx'=a2b2 is the equation to the tangent at P.

5. CT=22, CT'=b2.

y

b2

, PG being the normal at P.

6. CG e'x, GN=

CD2

7. QV2=CP2 '

PV.VP', where QV and CD are parallel to PT.

8. CP2+CD2-a2+b2 and CD2 SP.PH.

63. If A, S, C be the vertex, focus and centre of an ellipse; it is required to show that if SC become infinite, AS remaining finite, the ellipse will be changed into a parabola.

64. In an ellipse find the position of that focal distance SP which is a mean proportional to the semi-axes; when a=50, b=30.

65. If in any ellipse there be taken three abscissas in arithmetical progression; the radius-vectors drawn from the focus to the extremities of the ordinates at those points will also be in arithmetical progression.

For this question use formula (3).

66. In an ellipse whose semi-axes are 5 and 4, find the position of CP when an arithmetic mean between CA and CB.

67. At what point in an ellipse is the angle, formed by the two focal distances, greatest?

69. If PQ be two points in an ellipse, such that the lines CP,

CQ are at right angles to each other, then will
Formula (12) will here be of service.

I

p+

I

I I +

CP2 CQ2
CQ2=a2 b2

70. If CP be any semidiameter of an ellipse, and AQO be drawn from the extremity of the major axis parallel to CP, and meeting the curve in Q and the minor axis produced in O; show that 2CPAO.AQ.

Find the ordinates to Q and O, by means of the equations to the straight line AQO and to the ellipse; then by similar triangles express these ordinates in terms of AO, AQ and CP.

71. The length of the perpendicular upon the tangent from the centre of an ellipse is equal to a 1-e cos', where is the inclination of the tangent to the axis major.

72. If a be the acute angle between the tangent and focal distance at any point of an ellipse, the distance of that point from the centre is equal to a-b2 cotλ.

First show that CP2=a2+ e2x2-a2e2; then find a relation between

r and a from the equation

PK
PG

= sin λ (see the figure); this gives

b2 cot'λ=a2e2— e2x2.