The Elementary Geometry of Conics
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1888 edition. Excerpt: ... tan 6 =-.-.-. = i, therefore eie = 0 or oo; or i6 =t Oo; and therefore 6 + a or 0 is of the form a i. oo. In other words the anomalous equation, tan (0 + a) = tan 6 = i, signifies that the tangent of the complex angle a i. /? approaches asymptotically to the limit i as fi becomes infinite. A straight line through a Focoid is commonly said to be only at right angles to itself, because in the expression for the tangent of the angle between y = mx, and y = m'x, the denominator 1 + mm' vanishes when m = m-- i. But the numerator m m' also vanishes, and the angle is indeterminate. Art. 79. PROBLEMS. The General Conic. 1. The distance of any point outside a conic from the focus is to its perpendicular distance from the directrix in a ratio greater than the eccentricity, and conversely. 2. If an ellipse, a parabola, and a hyperbola have the same focus and directrix, the ellipse lies wholly within the parabola, and the parabola wholly within the hyperbola. 3. Conics having the same focus and directrix do not meet. 4. If SL be the semi-latus rectum, and SD be drawn parallel to PR (Art. 4) to meet the directrix, then SP: PR = SL: SD. 5. The segments of any focal chord subtend equal (or supplementary) angles at the foot of the directrix. 6. Determine the pole of the latus rectum and the polar of the focus. Art. 69. 7. If chords PR, QR be produced to meet the directrix in p, q, the angle between the focal radii to p, q will be equal (or supplementary) to half the angle between the focal radii to P, Q. 8. If two tangents TP, TQ meet any third tangent in p, q, the angle betweeu the focal radii to p, q will be equal (or supplementary) to half the angle between the focal radii to P, Q. 9. The lines joining the extremities of...
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