106. To express the perpendicular from the origin on the tangent at any point in terms of the radius vector of that point.

Let SQ be the perpendicular from the origin on the tangent at P, and suppose SQ=p; then

that is,

SC2= SP2+PC-2SP. PC cos SPC

= SP2+PC2 −2SP. PC sin SPQ;

In the figure S and C are on the same side of the tangent at P. If we take P so that the tangent at P falls between S and C, we shall find

107. These equations are sometimes useful in the solution of problems, or demonstration of properties of the circle. For example, take the equation (4) in Art. (105),

r2 - 2rl cos + l2 − c2 = 0 ;

by the theory of quadratic equations we see that the product of the two values of r corresponding to any value of is l-c3, which is independent of 0. This agrees with Euclid III. 35, 36.

Also the sum of the two values of r is 27 cos 0; hence if a line be drawn through the pole at an inclination to the initial line, the polar co-ordinates of the middle point of the chord which the circle cuts off from this line are

21 cos 0

2

and 0; that is, 7 cos 0, and 0.

Hence the polar equation to the locus of the middle point of the chord is

r = l cos 0,

which by (5) in Art. 105, is a circle, of which the diameter is l.

EXAMPLES.

1. Determine the position and magnitude of the circles

(1) x2+ y2+ 4y — 4x − 1 = 0,

(2) x2 + y2+6x-3y — 1 = 0.

2. Find the points of intersection of the circle

with the lines

y2+x2 = 25

y+x=-1, y+x=-5, and 3y+4x=-25.

3. A circle passes through the origin and intercepts lengths h and k respectively from the positive parts of the axes of x and y; determine the equation to the circle.

4. A circle passes through the points (h, k) and (h', k'); shew that its centre must lie on the line

5. On the line joining (x', y') and (x", y") as diameter a circle is described; find its equation.

6. A and B are two fixed points, and P a point such that AP=mBP, where m is a constant; shew that the locus of P is a circle, except when m = 1.

7. The locus of the point from which two given unequal circles subtend equal angles is a circle.

8. Find the equation which determines the points of intersection of the line

Deduce the relation that must hold in order that the line may touch the circle.

9. Find the equation to the tangent at the origin to the circle

10. Shew that the length of the common chord of the circles whose equations are

11. A point moves so that the sum of the squares of its distances from the four sides of a square is constant; shew that the locus of the point is a circle.

12. A point moves so that the sum of the squares of its distances from the sides of an equilateral triangle is constant; shew that the locus of the point is a circle.

13. A point moves so that the sum of the squares of its distances from any given number of fixed points is constant; shew that the locus is a circle.

14. Shew what the equation to the circle becomes when the origin is a point on the perimeter, and the axes are inclined at an angle of 120°, and the parts of them intercepted by the circle are h and k.

15. What must be the inclination of the axes that the equation

x2+ y2- xy-hx-hy = 0

may represent a circle? Determine the position and magnitude of the circle.

16. What must be the inclination of the axes that the equation

x2 + y2+ xy — hx – hy = 0

may represent a circle? Determine the position and magnitude of the circle.

17. Determine the equation to the circle which has its centre at the origin, and its radius = 3, the axes being inclined at an angle of 45°.

2

18. Determine the equation to the circle which has each of the co-ordinates of its centre - and its radius = the axes being inclined at an angle of 60o.

√3'

19. The axes being inclined at an angle o, find the radius of the circle

x2 + y2+ 2xy cos w-hx-ky = 0.

20. Shew that the equation to a circle of radius c referred to two tangents inclined at an angle w as axes is

21. Shew that the equation in the preceding question may also be written

22. Find the value of c in order that the circles

(x − a)2 + (y — b)2 = c3, and (x-b)2 + (y — a)2 = c2,

may touch each other.

23. ABC is an equilateral triangle; take A as origin, and AB as axis of x; find the rectangular equation to the circle which passes through A, B, C. Deduce the polar equation to this circle.

24. If the centre of a circle be the pole, shew that the polar equation to the chord of the circle which subtends an angle 28 at the centre is

where a is the angle between the initial line and the line from the centre which bisects the chord. Deduce the polar equation to a line touching the circle at a given point.

25. Find the polar equation to the circle, the origin being on the circumference and the initial line a tangent. Shew

that with this origin and initial line, the polar equation to the tangent at the point e' is

26. Shew that if the origin be on the circumference of a circle, and the diameter through that point make an angle a with the initial line, the equation to the circle is

r = 2c cos (0 − a).

27. Determine the locus of the equation

r = A cos (0 − a) + B cos (0 − ß) + C cos (0 − y) +

......

28. AB is a given straight line; through A two indefinite straight lines are drawn equally inclined to AB, and any circle passing through A and B meets those lines in L, M; shew that the sum of AL and AM is equal to a constant quantity when L and M are on opposite sides of AB, and that the difference of AL and AM is constant when L and M are on the same side of AB.

29. ABC is an equilateral triangle, and

find the locus of P.

PA = PB+ PC,

30. There are n given straight lines making with another fixed straight line angles a, B, Y, ......; a point P is taken such that the sum of the squares of the perpendiculars from it on these n lines is constant; find the conditions that the locus of P may be a circle.

31. A point moves so that the sum of the squares of its distances from the sides of a regular polygon is constant; shew that the locus of the point is a circle.

32. A line moves so that the sum of the perpendiculars AP, BQ, from the fixed points A and B is constant; find the locus of the middle point of PQ.

33. O is a fixed point and AB a fixed line; a line is drawn from O meeting AB in P; in OP a point is taken so that OP. OQ=k; find the locus of Q.