123. The Parabola.-Given the equation

y2 — 2xy + x2-4y+x−2=0,

(1)

for which b2-4ac=0, to find its locus. Solving for y, we get

As with the ellipse,

y=x+2±√3(x+2).

y=x+2

(2)

(3)

is a diameter bisecting vertical chords.

If x is less than -2, y is imaginary, so that there are no points of the curve to the left of x=-2, which is the vertical limiting tangent. As x increases, √3(x+2), the length of the halfchord, increases without limit, remaining real, so that the curve is open, spreading indefinitely.

If we put x=1, we find

y=1+2±√3(1+2)=3±3=6 or 0.

The points (1, 6) and (1, 0) thus lie on the parabola. Any number of points may be located in this way, but we have enough to trace the curve, as in Fig. 23.

124. Tracing a Parabola.-The steps necessary in tracing a parabola may be summarized:

Given an equation

Ax2+Bxy+Cy2+Dx+Ey+F=0,

for which B2-4AC=0:

(1) Solve for y in terms of x.

(2) Trace the oblique diameter.

(3) Draw the vertical limiting tangent.

(4) Compute y for some one value of x, and plot the corresponding points.

(5) Sketch a parabola through these two points and tangent to the vertical limiting tangent where it crosses the oblique diameter.

125. Diameters of the Parabola.-The equation of a parabola can always be solved in a variety of ways; for instance, besides solving the equation of Art. 123 for y, we can solve for x in terms of y, getting

x=y−1±√12y+9,

showing that the parabola is intersected by the diameter y=x+1 where this diameter is crossed by the horizontal limiting tangent y= -2.

Again, we can write immediately

y_x=±√4y−x+2,

showing that the parabola is intersected by the diameter y=x where this diameter is crossed by the oblique limiting tangent 4y-x+2=0.

126. Construction of the Parabola.-It can be shown that any number of points may be located on a parabola as follows: Given a parallelogram KK'BC, Fig. 24, bisected by a line which passes through A, the middle point of KK', and A', the middle point of CB. Lay off any lengths along AA' and number the points of division consecutively. Lay off the same

FIG. 24.

lengths, correspondingly numbered, along K'B and KC. Join C with any point of AA', and A with the corresponding point of K'B; the two lines will meet at a point P of the parabola tangent to KK' at A, passing through B and C, and having AA' as the diameter that bisects chords parallel to KK'.

Obviously lines from C and A give only half the parabola; lines from B to points of AA' and from A to corresponding points of KC give the other half.

Examples.

1. Trace x2+2xy + y2-x-4y+3=0, finding also the slope where it crosses the y-axis.

2. Trace 4x2+4xy+ y2+4y-8x-5=0, finding the slopes where it crosses the axes.

3. At any point of the parabola of Example 2,

write the condition that this direction shall be perpendicular to that of the diameters. What is the slope of the line represented by this equation? At what point does this line cut the parabola? What is the line?

4. Find the locus of points equally distant from y+x=1 and the point (1, 1), and trace the curve.

5. Given a point P(2, 3) and two straight lines (1) 2y=x and (2) y=3x, find and trace the two loci (a) of points equidistant from P and (1); and (b) of points equidistant from P and (2). Thus locate the points equidistant from P, (1) and (2).

127. The Hyperbola.-In the equation whose second-degree terms may be resolved into two unequal factors, there are peculiar properties of its locus which require special consideration. Suppose the simplest form of such an equation be taken,

where the linear factors of the second-degree term are x and y. Solving for y, and then for x, we obtain

Since x and y can be interchanged without changing the equation, we infer that the curve is symmetrical to the line y=x (the dotted line in the figure) and the point (1, 1) is on the curve. Beginning with this point, arrange the corresponding values of x and y in series:

It is evident that as x increases indefinitely, y decreases indefinitely; i. e., the curve approaches indefinitely near the x-axis; or in symbols, when x is ∞o, y is 0, and the curve touches the x-axis at infinity. It may be shown in the same way that the curve is tangent to (touches) the axis of x at∞o, and from equation (3), that the y-axis is also tangent at ±∞.

FIG. 25.

The curve is called a hyperbola, and the two lines having the peculiar property of approaching indefinitely close to the curve as the curve becomes indefinitely distant from the origin are called asymptotes.

If we find the intersections with the curve of any straight line through the origin, y=mx, we obtain the points

1

1

(√'m, √m), (− √'m, −√m);

that is, every chord through the origin is bisected at the origin, which is therefore the center of the curve, and such a chord is a diameter.