...ylc—2a(x+h). (Arts. 86 — 90, 92, 100 may be slightly altered so as to suit the parabola.) 113. A diameter of a curve is the locus of the middle points of a system of parallel chords. For example, in the circle a diameter is, according to this definition,...

...ellipse into an ellipse, a parabola into a parabola, a hyperbola into a hyperbola. A diameter, being the locus of the middle points of a series of parallel chords, projects into a diameter, and therefore the centre of a conic (being the intersection of two diameters)...

...touching it at a given point. 4. In an ellipse referred to its centre as origin, find the equation to the locus of the middle points of a series of parallel chords. What are conjugate diameters ? 5. If S and H are foci of a hyperbola, and P any point of it, assuming...

...particular cases of the locus, due to bringing the point (x' , y') on the curve, or outside of it. 62. A Diameter of a curve is the locus of the middle points of parallel chords: To find the, equation of any diameter. Let (.?;, y) be the middle point P of the chord...

...:: 2AN : PN :: PN : 2AS ; Therefore the line through P bisecting Q(£ is parallel to the axis. Hence the locus of the middle points of a series of parallel chords is a straight line parallel to the axis. 68 1 . Take CP, CQ two tangents such that PCQ is twothirds...

...at the extremities of a chord whose length is constant. THE CIRCLE. SUPPLEMENTARY PROPOSITIONS. 68. A Diameter of a curve is the locus of the middle points of a system of parallel chords. The chords which any diameter bisects are called the Chords of that diameter....

...the squares of the radii g* + f* — c, gn +/'2 — c, respectively, the condition is or 60. To find the locus of the middle points of a series of parallel chords of a circle. Let x'y' be the coordinates of any point P, xy of any point Q on the circle ,*•"+ y*+2gx...

...given distance from a given circumference ? a radius distance from a given circumference ? 162. What is the locus of the middle points of a series of parallel chords in a circle ? 163. What is the locus of the middle points of equal chords in a circle ? 164. What is...

...pts. ? Conclusion. 125. Cor. Two circles cannot intersect in more than two points. EXERCISES 76. What is the locus of the middle points of a series of parallel chords ? 126. The ratio of one quantity to another of the same kind is the quotient of the first divided by...

...T,xy +y2 + x +y + 1 = o ? Ascertain whether the others cut the curve in real points. 12 8. Prove lhat the locus of the middle points of a series of parallel chords of an ellipse is a straight line. Prove that the tangents at the points where this locus cuts the curve...