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49. If the two sides CA, CB of a spherical triangle CAB 1835 be quadrants, shew that C is the pole of the arc AB.
50. In applying Napier's rules to the solution of quadrantal spherical triangles, what are to be considered the circular parts? Having given the two sides, find the angle opposite to the quadrantal side.
51. Express the cotangent of the angle of a spherical triangle in terms of another angle and the sides including it, and reduce the formula to one adapted to logarithmic computation.
52. If in a right-angled spherical triangle a, ß be the arcs drawn from the right angle respectively perpendicular to, and bisecting the hypothenuse c, shew that
sin ß sin ,
VI + (sin a)? determine also the cosine of a side in terms of a and ß.
53. In a spherical triangle, determine the limits of the sum of the angles. Prove also that the difference between any angle and the sum of the other two is less than 180°.
54. If one angle and the hypothenuse of a right-angled spherical triangle be given, shew that there is no ambiguity in determining the other parts. In what case is the solution of a right-angled triangle ambiguous ?
55. The three angles are not sufficient data for the determination of a plane triangle; explain why they are sufficient in a spherical triangle.
In a spherical triangle having given two sides and an angle opposite to one of them, determine the other parts.
56. The arc of a great circle, which joins the middle point of a side of a spherical triangle with its pole, bisects the angle formed by the arcs which join the pole with the poles of the other sides.
57. If one angle of a triangle, either plane or spherical, be equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.
QUESTIONS IN ANALYTICAL GEOMETRY.
1. Find the equation of a straight line, which shall pass through two points whose coordinates are given.
2. Given the base of a plane triangle and the difference of the angles at the base, to find the curve traced by the vertex.
3. If the area of a curve between any two values of one abscissa can be expressed in finite terms, shew that the area between two values of any other abscissa of the same curve can be found,
4. A normal drawn to a cissoid at the point where it cuts the generating circle, meets the axis produced in a certain point; prove that the line intercepted between this point and the vertex of the cissoid is divided into three equal parts by the centre and the further extremity of the diameter, of the generating circle.
5. A ship (P) begins to sail towards another (B) from a fixed point (C). At the same instant B begins to move in a direction perpendicular to P's first motion : P is always found in the line joining C and B; but can only accelerate her rate of sailing so as to retain the same distance from B as at first. What is the curve traced by P?
6. AN and NP are the abscissa and ordinate of a cissoid, the diameter of whose generating circle is AP: AP is joined and NQ always taken equal to it. Prove that the whole area of the curve traced out by Q : AB2 :: 4:3.
7. Inscribe a semicircle in a quadrant.
8. If a, ß, y .... be the roots of the equation X = 0, and A, B, C .. the results when these roots are substituted for x in the limiting one; then will X X
X Y = at b +
C+ A (2 – a) B(x – ) C (x - y) be the equation of the parabolic curve which passes through the points of which a, ß, y... are the abscissas, and a, b,c... the corresponding ordinates.
9. If two lines SP, HP revolve about the points S, H, so that SP x HP = CS?, (C being the middle point of SH) then the locus of the point P is the Lemniscate of Bernoulli.
10. Describe a circle about a given segment of a parabola made by an ordinate perpendicular to the axis.
11. Given the hypothenuse of a right-angled triangle and 1825 the side of an inscribed square. Required the two sides of the triangle.
12. Determine the locus of a point so situated within a plane 1826 triangle, that the sum of the squares of the straight lines drawn from it to the angular points is constant ; if the curve has a centre, determine its position.
13. What are the lines traced by the vertex and the focus 1827 of a parabola rolling on another equal to it, the vertices coinciding in one position ?
14. Explain the method of Geometrical Analysis, and by it solve the problem. In a given square to inscribe another square having its side equal to a given straight line. To what limitation is this line subject?
15. Draw a straight line touching a circle at a given point, without any other instruments besides a parallel ruler and a pencil.
16. If any two circles, the centres of which are given, inter- 1828 sect each other, the greatest line which can be drawn through either point of intersection and terminated by the circles is independent of the diameters of the circles.
17. Give a construction depending upon the cycloid, for determining an arc equal to its cosine.
18. Find the cosine of the angle contained between two straight lines whose equations are y = ax + b, and
= d'r +6. 19. Find the equation to the curve in which the distance of any point from a given fixed point is equal to the perpendicular drawn from the same point in the curve upon a given line.
20. Having given the equation to a straight line, find the equation to another straight line drawn perpendicular to it from a given point ; find also the length of the perpendicular.
21. Find the rectangular equation to the conchoid of Nicomedes, and draw a tangent to the curve.
22. If from two fixed points in the circumference of a circle, straight lines be drawn intercepting a given arc and meeting without the circle, the locus of their intersection is a circle.
23. Given the equations of two straight lines, find the equation to a third which shall pass through their point of intersection and make equal angles with them; and shew from the result that there are two straight lines at right angles to each other which satisfy the question. 24. In the general equation of the second degree
ay? + bxy + cx? + ex + fy + g = 0, shew in what cases the curve will be an ellipse, hyperbola and parabola ; and find the coordinates of the centre in the former
25. If a curve have as many asymptotes as it has dimensions, and a right line be drawn which cuts them all, the parts of the line measured from the asymptotes to the curve will together be equal to the parts measured in the same direction from the curve to the asymptotes.
26. Find the equation to a straight line passing through a given point, and cutting a given straight line at a given angle. Required also the coordinates of the point of intersection.
27. If y = ax + b, and y = d'x + b', be the equations to two straight lines in the same plane ; prove that the cosine of the angle contained between them is equal to
1 + an
V(1 + a2)(1 + a'?) 28. A straight line revolving in its own plane about a given point intersects a curve line in two points; find the curve when the rectangle of the lines intercepted between the given point and the points of intersection is constant.
29. Two straight lines, which are always taugents 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.
30. Find the magnitudes and positions of the principal axes of the curve of the second order, the equation of which is
Ay2 + Bxy + Cx2.4 Dy + Ex + F = 0. 31. x,y and X, Y are the coordinates of a point referred to two systems of rectangular coordinates having a common origin and inclined to each other at a given angle: find the relation subsisting between x, y, X and Y.
32. Having given the equations to two straight lines in the same plane, find the tangent of the angle contained between them.
33. If a, b, c be the lengths of the chords of three arcs of a circle, which together make up a semi-circumference, and r the radius of the circle, then
473 (a? + 62 + c2) r abc = 0. 34. Transform the equation to a plane curve, from one system of coordinates to another inclined at a given angle to the former, and having a different origin, both systems being rectangular; and take the example ya = mx + nx?.
35. If a straight line be drawn from a given point perpendicular to a given straight line, find the coordinates of the point of intersection, and the length of the perpendicular.