Page images
PDF
EPUB

33. If two circles cut orthogonally, prove that an indefinite number of pairs of points can be found on their common diameter such that either point has the same polar with respect to one circle that the other has with respect to the other. Also shew that the distance between such pairs of points subtends a right angle at one of the points of intersection of the two circles.

34. If the equations of two circles whose radii are a, a' be S=0, S'= 0, then the circles

[merged small][ocr errors][merged small][merged small][merged small]

35. Find the locus of the point of intersection of two straight lines at right angles to one another, each of which touches one of the two circles

(x − a)2 + y2 = b2, (x+a)2 + y2 = c2,

[ocr errors]

and prove that the bisectors of the angles between the straight lines always touch one or other of two other fixed circles.

36. Shew that the diameter of the circle which cuts at right angles the three escribed circles of the triangle ABC is

a (1 + cos A cos B + cos B cos C+ cos C'cos 4)

sin A

37. Find the locus of the point of contact of two equal circles of constant radius c, each of which passes through one of two fixed points at a distance 2a apart and shew that, if a = c, the locus splits up into a circle of radius a and a curve whose equation may be put into the form (x2+y)2= a2 (x2-3y2).

A2 x + 2 H xy + By 2 + 2 4 x + 2y + C =

[ocr errors][merged small]

H2-AB <0 or my

[ocr errors]

жу

[blocks in formation]
[ocr errors][merged small][merged small][merged small]

89. Definitions. A Conic Section, or Conic, is the locus of a point which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line. The fixed point is called a focus, the fixed straight line is called a directrix, and the constant ratio is called the eccentricity.

It will be shewn hereafter [Art. 312] that if a right circular cone be cut by any plane, the section will be in all cases a conic as defined above. It was as sections of a cone that the properties of these curves were first investigated.

We proceed to find the equation and discuss some of the properties of the simplest of these curves, namely that in which the eccentricity is equal to unity. This curve is called a parabola.

90. To find the equation of a parabola.

Let S be the focus, and let YY' be the directrix. Draw SO perpendicular to YY', and let OS=2a. Take OS for the axis of x, and OY for the axis of y.

Let P be any point on the curve, and let the coordinates of P be x, y.

Draw PN, PM, perpendicular to the axes, as in the figure, and join SP.

[ocr errors]
[blocks in formation]

The curve cuts the axis of x at a point A where y=0 and from (i) when y = 0, x = a; that is, OA = a. The point A is called the vertex of the parabola.

If we transfer the origin to A, the axes being unchanged in direction, equation (i) will become [Art. 49]

Also

and

y2 = 4ax

..........(ii). The focus is the point (a, 0). The directrix is the line

x + a = 0.
SPMP=0A+AN=a+x.

91. Since the equation of the parabola is y=4ax, y2 is a positive quantity, a must always be positive,

and therefore the curve lies wholly on the positive side of the axis of y.

For any particular value of x there are clearly two values of y equal in magnitude, one being positive and the other negative. Hence all chords of the curve perpendicular to the axis of x are bisected by it, and the portions of the curve on the positive and on the negative sides of the axis of x are in all respects equal.

As x increases У also increases, and there is no limit to this increase of x and y, so that there is no limit to the curve on the positive side of the axis of y.

The line through the focus perpendicular to the directrix is called the axis of the parabola.

The chord through the focus perpendicular to the axis is called the latus-rectum.

In the figure to Art. 90, SL=LK = OS = 2a. Therefore the whole length of the latus-rectum is 4a.

92. We have found that y2-4ax = 0 for all points on the parabola.

For all points within the curve y2 - 4ax is negative.

For, if be such a point, and through Q a line be drawn perpendicular to the axis meeting the curve in P and the axis in N, then Q is nearer to the axis than P and therefore NQ' is less than NP2. But, P being on the curve, NP2 - 4a. AN=0, and therefore NQ-4a. AN is negative.

Similarly we may prove that for all points outside the curve y2-4ax is positive.

Hence, if the equation of a parabola be y2-4ax = 0, and we substitute the co-ordinates of any point in the lefthand member of the equation, the result will be positive if the point be outside the curve, negative if the point be within the curve, and zero if the point be upon the

curve.

93. The co-ordinates of the points common to the straight line, whose equation is y = mx + c, and the

parabola, whose equation is y2 = 4ax, must satisfy both equations.

Hence, at a common point, we have the relation,

[ocr errors]

.(i). Therefore the abscissæ of the common points are given by the equation (i), which may be written in the form m2x2 + (2mc - 4a) x + c2 = 0...... .(ii).

Since (ii) is a quadratic equation, we see that every straight line meets a parabola in two points, which may be real, coincident, or imaginary.

When m is very small, one root of the equation (ii) is very great; when m is equal to zero, one root is infinitely great. Hence every straight line parallel to the axis of a parabola meets the curve in one point at a finite distance, and in another at an infinite distance from the vertex.

94. To find the condition that the line y = mx + c may touch the parabola y2 — 4ax = 0.

As in the preceding Article, the abscissæ of the points common to the straight line and the parabola are given by the equation

that is

(mx + c)2 = 4ax,

m2x2 + (2mc - 4a) x + c2 = 0.

If the line be a tangent, that is if it cut the parabola in two coincident points, the roots of the equation must be equal to one another. The condition for this is 4m2c2 = (2mc-4a)”,

[blocks in formation]

will touch the parabola y2 - 4ax = 0.

95. To find the equation of the straight line passing through two given points on a parabola, and to find the equation of the tangent at any point.

« PreviousContinue »