EQUATION OF THE CIRCLE.

109

PROPOSITION VI.

To find the Equation of the Circle referred to Rectangular Coördinates.

T

y

X

Fig. 42.

Let OX, OY, be two lines, intersecting each other at right angles in O; also let o, situated at the distance b from OX and a from OY, be the centre of any circle; further, let P be any point of the circumference, distant from OX and OY by the variable lines and ; then will the radius, r, be the hypothenuse of a right angled triangle, of which the base will be x — a, and the perpendicular y — b, whence

y

The Equation of the Circle (y—b)2 + (x − a)2 = r2.

(189)

Scholium. The variables x, y, are called Coördinates of the point, P, or of the circumference, when spoken of together; when one is to be distinguished from the other, y is denominated the ordinate and r the abscissa; OX, OY, are called the axes of coordinates, OX is the axis of x, OY that of y-O is their origin. If the centre o of the circle be on OX, the axis of x, b = 0, and the equation (189) becomes

y2 + (x − a)2 = p2.

(190)

Fig. 422.

If, in addition to b=0, we make a = 0, the centre, o, will be transported to the origin, O, and the equation becomes

Resolving (191) in reference to y, we find two

equal values of y for every value of x,

=

Now, if we diminish the arc AD, its ordinate AB =y diminishes also, and becomes 0 when AD=0; finally AD reappearing as A'D measured in the opposite direction, its ordinate A'B reappears, likewise measured in the opposite direction from B, and

is thereforey, according to the principle laid down in (180). Whence,

Cor. 1. The circle is perfectly symmetrical, any diameter, (192) as DD',bisecting all the chords to which it is perpendicular[fig. 42,]. Cor. 2. The perpendicular which bisects a chord passes (193) through the centre of the circle and bisects the arc.

Cor. 3. The greater chord is less distant from the centre, (194) since the semichord y increases as a diminishes; and the diameter is, therefore, the greatest straight line that can be drawn in a circle. By comparing equations (189), (190), (191), with the figures (42), (422), (42), which illustrate them, it is obvious that the coordinates y, x, represent, in general, totally different quantities in these different forms, so that one cannot be combined with another, as in ordinary algebraical operations. But if the circles intersect, as in figure 424, then the point of intersection, I, being the same for both circumferences, the coördinates of this point, y, x, will satisfy the equations of both circles, and from (191) and (190) denoting the different radii by r, r2, we have

whence

Y;2 + x;2 = r2,

y2 + (x; − a)2 = r22 ;

2ax; — a2 = r2 — r22.

Resolving the last equation in reference to a, we find

whence, for any compatible values of x, r1⁄2, r, we find two values for the distance of the centres, a; or, for any given radii and a given chord, the circles can be made to intersect in two ways.

If r2 <r, the values of a will be both, the one greater than , the other less, and the centres will be

If we take x¡ instead of +x ̧, a = x; ±(x‚2+r‚2—r2)?,

becomes a = −x, ±(x;2+r‚2—r2)*, or — a = x; = (x‚2 + r ̧2 — r2)‡ ; whence we have the same constructions over again, only on the opposite side of the origin.

whence it follows that the circumferences intersect in two points, I, I', equally distant from the middle, B, of their common chord; and .. that, as long as y; has a real value, the triangle OL, must be possible. From what has been demonstrated, we have the following corollaries:

Cor. 4. Two circles of given radii will have for a com- (195) mon chord four positions of intersection.

Fig. 428.

Cor. 5. The line joining the centres of intersecting cir- (196) cles, is perpendicular to, and bisects their common

chords.

Fig. 429.

Cor. 6. In order that two circles intersect, of the three (197) quantities, their radii and the distance of their centres, any two must be greater than the third.

Cor. 7. The distance of the centres of two tangent cir- (198)

cles, is equal to the sum or difference of their radii, according as they touch externally or internally.

Fig. 4210

Cor. 8. The line joining the centres of tangent circles, (199) passes through the point of tangency, and the converse. [fig. 4210.]

EXERCISES.

10. What is the greatest triangle that can be inscribed in a semicircle? Ans. An isosceles triangle, standing upon the diameter. 2o. What is the greatest rectangle that can be inscribed in a given circle? Ans. A square. 3°. What is the maximum triangle that can be inscribed in a given circle and standing on its radius?

Ans. A triangle right angled at the centre.

4°. How many boards 15 inches wide and an inch thick can be cut from a log 20 inches in diameter? Ans. 5 √7.

5°. What must be the diameter of a water wheel to fit an apron 2a feet across and b feet deep?

6o. How much must a plank be cut out to make a felly 1 ft. 6 in. long to a wheel 6 ft. in diameter, the measures being taken on the inside?

7°. What is the radius of the largest circle that can be cut from a triangular plate of silver, measuring 23, 3, 3'5 inches on its sides?

8°. Three brothers, residing at the several distances of 10, 11, 12 chains from each other, are to dig a well which shall be equally distant from them all. What must be that distance?

9o. The diameters of the fore and hind wheels of a carriage are 4 and 5 feet, and the distance of their centres 6 feet. At what point will a line joining these centres intersect the ground, supposed to be a plane?

10°. There are two wheels situated in the same vertical plane, and their centres in the same vertical line; the largest, the centre of which is 10 feet below the floor, is 8 feet in diameter, and the smaller, the centre of which is 6 feet above the floor, is one foot in diameter. Where must we cut through the floor for the passage of the strap that is to embrace the wheels?

11°. The same as in the above, except that the strap is to cross between the wheels.

12°. The same conditions as in 10°, except that the centre of the lower wheel is 4 feet from the plumb line dropped from the centre of the wheel above the floor.

13°. It is required to construct three equal friction wheels to run tangent to each other and to an axle two inches in diameter. What must be their common radius, and what the radius of the circular bed cut for them in the centre of a wheel?

14°. If the top-masts of two ships, having elevations of 90 and 100 feet above the level of the sea, are seen from each other at the distance of 25'7 miles, what is the diameter of the earth?

15°. How far can the Peak of Teneriffe be seen at sea?

16°. How far will a water level fall away from a horizontal line, sighted at one end in a distance of one mile, the diameter of the earth being estimated at 7,960 miles?

170.* If AC, one of the sides of an equilateral triangle ABC, be produced to E, so that CE shall be equal to AC; and if EB be drawn and produced till it meets in D, a line drawn from A at right angles to AC; then DB will be equal to the radius of the circle described about the triangle.

18°. If an angle B of any triangle ABC, be bisected by the straight line BD, which also cuts the side AC in D, and if from the centre A with the radius AD, a circle be described, cutting BD or BD produced in E; then BE: BD :: AD : CD.

19°. Let ABC be a triangle right angled at B; from A draw AD parallel to BC, and meeting in D, a line drawn from B at right angles to AC; about the triangle ADC describe a circle, and let E be the point in which its circumference cuts the line AB or AB produced; then AD, AB, BC, AE, are in continued proportion.

20°. Let ABC be a circle, whose diameter is AB; and from D any point in AB produced, draw DC touching the circle in C, and DEF any line cutting it in E and F; again, draw from C a perpendicular to AB, cutting EF in H; then,

ED2 CD2 :: EH : FH.

21°. Let ABC be a circle, and from D, a point without it, let three straight lines be drawn in the following manner: DA touching the circle in A, DBC cutting it in B and C, and DEF cutting it in E and F; bisect the chord BC in H, draw AH, and produce it till it meets the circumference in K; draw also KE and KF cutting BC in G and L. The lines HG and HL are equal.

* "Prize Problems," Yale College, 1840