EXAMPLES.

=

1. Change the equation ra2 cos 20 into one between x and y.

2. Shew that the equation 4xy-3x2=a2 is changed into x3 — 4y2 = a2, if the axes be turned through an angle whose tangent is 2.

3. Transform √x + √y = √c so that the new axis of x may be inclined at 45° to the original axis.

4. The equation to a curve referred to rectangular axes is +4ay cot a -4ax = 0; find its equation referred to oblique axes inclined at an angle a retaining the same axis of x.

5. Shew that the equation x2y2= a (x3+y3) will admit of solution with respect to y' if the axes be moved through an angle of 45°.

6. If x, y be co-ordinates of a point referred to one system of oblique axes, and x', y' the co-ordinates of the same point referred to another system of oblique axes, and

CHAPTER VI.

THE CIRCLE.

88. WE now proceed to the consideration of the loci represented by equations of the second degree; the simplest of these is the circle, with which we shall commence.

To find the equation to the circle referred to any rectangular

axes.

D

N

M

X

Let C be the centre of the circle; P any point on its circumference. Let c be the radius of the circle; a, b the coordinates of C; x, y the co-ordinates of P. Draw CN, PM parallel to OY, and CQ parallel to OX. Then

that is,

or

x2 + y2 — 2ax — 2by + a2 + b2 — c2 = 0 ......................... (2).

This is the equation required.

The following varieties occur in the equation.

I. Suppose the origin of co-ordinates at the centre of the circle; then a=0, and b=0; thus (1) and (2) become

II. Suppose the origin on the circumference of the circle; then the values x = 0, y = 0, must satisfy (1) and (2); therefore

which relation is also obvious from the figure, when O is on the circumference; hence (2) becomes

III. Suppose the origin is on the circumference, and that the diameter which passes through the origin is taken for the axis of x; then b=0, and a2=c2; hence (2) becomes

2

Similarly if the origin be on the circumference and the axis of y coincide with the diameter through the origin, we have a = 0, and b2 = c2; hence (2) becomes

Hence we conclude from (2) and the following equations, that the equation to a circle when the axes are rectangular is always of the form

x2+ y2+ Ax + By + C = 0,

where A, B, C are constant quantities any one or more of which in particular cases may be equal to zero.

89. We shall next examine, conversely, if the equation x2 + y2+ Ax + By + C = 0.......

(1)

I. If A

B2-4C be negative, the locus is impossible.

II. If A2+ B2-4C=0, equation (2) represents a point

considered as a circle which has an indefinitely small radius.

III. If A2 + B2 −4C be positive, we see by comparing equation (2) with equation (1) of the preceding article that it represents a circle, such that the co-ordinates of its centre are B and its radius (42 + B2 − 4 C′ )3.

A

2

و.

2

It will be a useful exercise to construct the circles represented by given equations of the form

or

x2 + y2+ Ax + By + C = 0.

For example, suppose

x2 + y2+4x-8y — 5=0,

x + 2)2+(y-4)=5+4+16=25.

Here the co-ordinates of the centre are 2, 4, and the radius is 5.

Tangent and Normal to a Circle.

90. DEF. Let two points be taken on a curve and a secant drawn through them; let the first point remain fixed and the second point move on the curve up to the first; the secant in its limiting position is called the tangent to the curve at the first point.

91. To find the equation to the tangent at any point of a circle.

Let the equation to the circle be

Let x', y' be the co-ordinates of the point on the circle at which the tangent is drawn; and x", y' the co-ordinates of

an adjacent point on the circle. The equation to the secant through (x', y') and (x", y') is

Now since (x', y') and (x", y') are both on the circumfer

ence of the circle,

.. by subtraction,

or

(x” — x′) (x' + x′) + (y" — y') (y′′ + y') = 0;

Now in the limit when (x", y") coincides with (x', y'), we have x" x', and y" y'; hence (3) becomes

=

Thus the equation to the tangent at the point (x', y') is

y-y':

=

(x-x')..

y

.(4).

This equation may be simplified; by multiplying by y' and transposing we have

xx' + yy' = x2 + y22 ;

12

.. xx' + yy' = c2...

.(5).

92. The equation to the tangent can be conveniently expressed in terms of the tangent of the angle which the line