CHAPTER III.

CHANGE OF AXES. ANHARMONIC RATIOS, OR CROSS

RATIOS.

INVOLUTION.

CHANGE OF AXES.

48. When we know the equation of a curve referred to one set of axes, we can deduce the equation referred to another set of axes.

49. To change the origin of co-ordinates without changing the direction of the axes.

Let OX, OY be the original axes; O'X', O'Y' the new axes; O'X' being parallel to OX, and O'Y' being parallel to OY. Let h, k be the co-ordinates of O' referred to the original axes.

Let P be any point whose co-ordinates referred to the old axes are x, y, and referred to the new axes x', y'. Draw PM parallel to OY, cutting OX in M and O'X' in N. Then

x= OM OK+KM=0K+ O'N=h+x',

=

y=MP=MN+NP=KO′+NP=k+y'.

Hence the old co-ordinates of any point are found in terms of the new co-ordinates; and if these values be substituted in the given equation, the new equation of the curve will be obtained.

In the above the axes may be rectangular or oblique.

50. To change the direction of the axes without changing the origin, both systems being rectangular.

Let OX, OY be the original axes; OX', OY the new axes; and let the angle XOX' = 0.

Let P be any point whose co-ordinates are x, y referred to the original axes, and x, y referred to the new Draw PN perpendicular to OX, PN' perpendicular to OX', N'M perpendicular to OX, and N'L perpendicular to PN, as in the figure.

axes.

Hence the old co-ordinates of any point are found in terms of the new co-ordinates; and if these values be substituted in the given equation, the new equation of the curve will be obtained.

Ex. 1. What does the equation 3x2 + 2xy + 3y2 – 18x - 22y+50=0 become when referred to rectangular axes through the point (2, 3), the new axis of x making an angle of 45° with the old?

First change the origin, by putting x' +2, y'+3 for x, y respectively. The new equation will be

3 (x' + 2)2 + 2 (x' + 2) (y' +3) +3 (y' + 3)2 – 18 (x' + 2) − 22 (y' +3) +50=0; which reduces to 3x2+2x'y' + 3y'2-1=0,

or, suppressing the accents, to

3x2+2xy + 3y2=1..........

To turn the axes through an angle of 45° we must write x'

which reduces to 4x2+2y'2=1.

Ex. 2. What does the equation x2 - y2+2x+4y=0 become when the origin is transferred to the point (− 1, 2)?

Ans. x2-y2+3=0.

Ex. 3. Shew that the equation 6x2+5xy - 6y2-17x+7y+5=0, when referred to axes through a certain point parallel to the original axes will become 6x2+5xy — 6y2=0.

Ex. 4. What does the equation 4x2+2√3xy+2y2-1-0 become when the axes are turned through an angle of 30°? Ans. 5x2+y2-1=0.

Ex. 5. Transform the equation x2-2xy + y2+x-3y=0 to axes through the point (-1, 0) parallel to the lines bisecting the angles between the original axes. Ans. √2y2-x=0.

Ex. 6. Transform the equation x2 + cxy+ y2 = a2, by turning the rectangular axes through the angle

π

51. To change from one set of oblique axes to another, without changing the origin.

S. C. S.

4

Let OX, OY be the original axes inclined at an angle w; and OX, OY' be the new axes inclined at an angle w'; and let the angle XOX' = 0.

VA

M

M

H

K

G

Let P be any point whose co-ordinates are x, y referred to the original axes, and x, y referred to the new axes; so that in the figure OM=x, MP=y, OM' = x', and M'P=y', MP being parallel to OY and M'P parallel to OY'.

Draw PK and M'H perpendicular to OX, and M'G perpendicular to PK.

Similarly, by drawing PL perpendicular to OY, we can shew that

xsin ∞ = x' sin X'OY — y' sin YOY'

=x'sin (-)-y' sin (w' +0-w).

These formulæ are very rarely used. The results which would be obtained by the change of axes are generally found in an indirect manner, as in the following Article.

*52. If by any change of axes ax2+2hxy + by2 be changed into a'x2 + 2h'x'y' + b'y2, then will

where w and w' are the angles of inclination of the two sets of axes.

If O be the origin and P be any point whose co-ordinates are x, y referred to the old axes and x', y' referred to the new, then OP2 is equal to x2 + y2+2xy cos w, and also equal to x2 + y2 + 2x'yˆcos w’.

Hence x2+ y2+ 2xy cos w is changed into

12

x22+y22+ 2x'y' cos w'.

Also, by supposition,

ax2 +2hxy + by3 is changed into a'x22 + 2h'x'y' + b'y'2. Therefore, if λ be any constant,

ax2+2hxy + by2+λ (x2 + 2xy cos w + y2) will be changed into a'x2 + 2h'x'y' + b'y'2 +λ (x22 + 2x'y' cos w' + y2). Therefore, if λ be so chosen that one of these expressions is a perfect square, the other will also be a perfect square for the same value of λ.

The first will be a perfect square if

(a + λ) (b + λ) − (h + λ cos w)2 = 0,

and the second if

(a' +λ) (b' + λ) — (h' +λ cos w')2 = 0.

These two quadratic equations for finding λ must have the same roots. Writing them in the forms