18. The diagonals of the three rectangular faces of a rectangular parallelopiped, which meet in a vertex of the solid, are respectively a, b, c. What is the area of the three faces?

19. If the first digit of a number containing 6 figures is interchanged with the fourth digit, the second with the fifth, the third with the sixth, a number is formed which, multiplied by the given number, is 122,448,734, 694, and which, diminished by the first, gives a remainder which is five times the first number. What is the number?

20. Calculate the sides of a right-angled triangle, given the perimeter 2p of the triangle and that the volumes of the solids generated by revolving the right-angled triangle about the two legs of the triangle is one half of the volume of the sphere whose radius is r. Suggestion.-Let x and y be the legs and z the hypotenuse of the triangle; then the equations will be

x + y + 2 = 2 p
xy(x+y)=2 μ3

x2+ y2 = 22, etc.

21. Find the four terms of a proportion, given that the sum of the extremes is 21, that of the means 19, and the sum of the squares of the four terms is 442.

22. Find the sides of a right-angled triangle, given the altitude h (hypotenuse being the base), and the difference, c, between the legs.

23. If the sum of any two of three numbers is multiplied successively by the third, the successive products are respectively 810, 680, and 512. What are the numbers?

24. Four quantities are in a proportion. The product of the extremes is a, the sum of the first two terms is b, and the sum of the last two terms is c. What are the four terins?

25. Four quantities are in a proportion. The sum of the first and fourth is a; the sum of the second and third is b; and the sum of the squares of the four quantities is c. What are the numbers?

CHAPTER XII

GRAPHICAL REPRESENTATION OF THE SOLUTIONS OF SYSTEMS OF SIMULTANEOUS QUADRATIC EQUATIONS

455. Graph of the General Quadratic Function ax2 + bx + c.

This problem is illustrated by the following example: EXAMPLE. Plot the equation y = x2 — 4 x — 5.

In the table below are arranged the various values of y which correspond respectively to values x = 0, + 1, + 2,

— 1, — 2, — 3, etc., in the equation y = x2 — 4

x ·

- 5.

.

X

5

2

0

+6

+7

etc.

0

+7

+16

NOTE 1.-It is clear that in case the values of x increase beyond those given in the table, each corresponding value of y will be larger than the one preceding; hence the values of x and y given in the table are sufficient to determine the ultimate directions of the curve. In general, when this is found to be the case, one need not compute more values for the table.

NOTE 2.-If the graph of an equation of the second degree in two variables consists of a single branch which extends to a part of the plane at an infinite distance from the origin, it is called a parabola.

Graphs of curves of the quadratic form

Ax2+2 Bxy + Cy2+ 2 D.x + 2 Ey + F= 0,

of the straight line ax + by + c = 0 (8245), and of the location of the points determined by their solutions.

EXAMPLE.-Plot the curves represented by the equations

and the points (243) represented by the solutions of these equations.

Plotting the points in the first table gives the line BAPST, Fig. 2. Plotting the points in the second table gives the curve R'Q'P'APQR. Plotting the points in the third table gives the eurve on the left, LA'M, which is in every respect equal to the curve on the right. The curves LA'M and R'AR are called the branches of the graph represented by the equation y2 = x2 — 16. *

*When the graph of a curve of the second degree in two variables consists of two branches, each of which extends to infinity, the graph is called an hyperbola. For example, y2x2-16 is the equation of an hyperbola (Fig. 2).

In reckoning the values of y which correspond to the values of x, in case of the hyperbola y2 = x2-16, we notice that for one value of x there correspond two values of y which are equal and opposite in sign. The same is true for the values of x which correspond to a value of y. For this reason the hyperbola is said to be symmetrical to the x- and y- axis.

which are represented by the points A and P respectively.

NOTE.-The points corresponding to the imaginary results x = 0, y = ±√-16, etc., are not situated on the hyperbola.

457. TYPE II.

(2442.)

r

Graphs of curves of the quadratic forms ax2 + bxy+cy2 = d and Ax2 + Bxy + Cy2 = D,

and of the location of the points determined by their solutions.

FIGURE 3

Plotting the points corresponding to the plus values of x and both plus and minus values of y in the second table, we get the branch curve BPAQB'; similarly, for the minus values of x and the corre

Y'

0

=±4.36··

8

V 11

sponding plus and minus values of y we get the branch curve BSA'RB'. These two branches make up the entire curve BAB'A' B which is the graph of the first equation.

Similarly, by plotting the values of x and y in the first table, we get the graph of the second equation, the hyperbola whose branches are PAQ and S.'R. The points corresponding to the imaginary values of x and y are not points on either of these curves. The solutions of this system of simultaneous quadratic equations are