...equilateral triangle, a circle, and a square are each 198, find the area of each figure. LOCI THEOREMS 1. The locus of the middle points of a system of parallel chords is a diameter perpendicular to the chords. 2. If two circles are tangent externally, their common internal...

...outside the limits of this work. 138. Definition and equation of a diameter. A diameter of a conic is the locus of the middle points of a system of parallel chords. Let Aa? + 2 Hxy + By*+ 2 Gx 0 (1) be any conic (fig. 140), ftS any chord which makes the angle 6 with...

...the middle point of the chord, then a'2mc 62c 14. From the preceding example, by eliminating c, prove that the locus of the middle points of a system of parallel chords, with slope m, of the ellipse is the straight line V— J ma2 This line is called a diameter of the...

...of rotation of a line about a point, into the definition of an angle, and "regards the diameter as the locus of the .middle points of a system of parallel chords, and the tangent as the limiting position of a secant. • It will thus be seen that Euclid's method,...

...lines are called conjugate diameters. 16. Prove that in the hyperbola — — — = 1 the equation of the locus of the middle points of a system of parallel chords of slope »» is 17. Through the point (z0, Jto) on the ellipse 62x2 + а2у2 = а362 a diameter is...

...the middle point of the chord, then a'2mc We 14. From the preceding example, by eliminating c, prove that the locus of the middle points of a system of parallel chords, with slope m, of the ellipse is the straight line 62 y- — ,xma2 This line is called a diameter of...

...of ;/=sin x; and write down tin- general 10 value of the angle, the sine of which ia J. GBOUP C. 1. The locus of the middle points of a .system of parallel chords 10 of a parabola is a straight line parallel to the axis of the parabola. Show that this line meets...

...major auxiliary circle. 122. Diameters. As already shown in Art. 108, the definition of a diameter as the locus of the middle points of a system of parallel chords leads directly to its equation, for the ellipse : viz., Eq. [52] 62 y = - -T- *• a2m The form of...

...a constant cross-ratio. 2. Assuming a focus to be an intersection of two isotropic tangents, prove that the locus of the middle points of a system of parallel chords of a conic is a straight line meeting the directrix on the straight line through the focus perpendicular...

...of a circle bisect a chord which does not pass through the centre, it cuts it at right angles. Find the locus of the middle points of a system of parallel chords drawn in a circle. (14) C 8. Show how to draw a circle which shall pass through two given points and'have...