...2 , &c., = «.AO 2 + J.BO 2 + c.CO 2 + <Z.DO 2 + &c., + (a + b + c + <Z, &c.) OP 2 . Cor. 1.—The locus of a point, the sum of the squares of whose distances from any number of given points, multiplied respectively by any system of constants a, J, c, d, is a circle, whose centre...

...if 6 = 0, c = 0, the question is satisfied by p = aa, whatever be a, therefore &c. Ex. 5. Find tJif locus of a, point, the sum of the squares of whose distances from a number of given planes is constant. Let S&,p1 = Ci, *S82pa = (7,, &c. be the equations of the given...

...distances from the four- sides of a square is constant ; shew that the locus of the point is a circle. 3. The locus of a point, the sum of the squares of whose distances from n fixed points is constant, is a circle. 4. A, B are two fixed points, and P moves so that PA = TO...

...gives the required equation (x - Xl) (x - x>) + (y - 2/1) (y - yji + (z - Zi) (z - z*) = 0. Ex. 2. The locus of a point, the sum of the squares of whose distances from any number of given points is constant, is a sphere. Ex. 3. A point moves so that the sum of the squares of its distances...

...lines gives the required equation (x - xj (x - x^j + (y - 2/0 (y - J/2) + (z - za) (z - z2) = 0. Ex. 2. The locus of a point, the sum of the squares of whose distances from any number of given points is constant, is a sphere. Ex. 3. A point moves so that the sum of the squares of its distances...

...rf.D02 + &c., + (a + b + o + d, &c.) OP2. This Proposition evidently includes the last. Cor. 1. — The locus of a point, the sum of the squares of whose distances from any numher of given points, multiplied respectively by any system of constants a, b, c, d, is a circle,...

...51. Locus of a point whose distances from two fixed points, A, B, are in a constant ratio m : n. 52. Locus of a point, the sum of the squares of whose distances from two fixed points, A, B, is constant, and equal to k2. 53. Locus of a point, the difference of the squares...

...d . DO2 + &c., + (0 + j + 0 + (^ &C.)OP2. This Proposition evidently includes the last. Cor. 1. — The locus of a point, the sum of the squares of whose distances from any number of given points, multiplied respectively by any system of constants a, b, c, d, is a circle, whose centre...

...distances from the four sides of a given square is constant ; show that the locus is a. circle. 3. The locus of a point, the sum of the squares of whose distances from n fixed points is constant, is a circle. 4. Find the locus of a point whose distances from two fixed...

...to twice the sum of the squares on half the line and on the part between the points of section. Find the locus of a point the sum of the squares of whose distances from two given points is constant. ' 5. Prove that a line perpendicular to a radius of a circle at its extremity...