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Conic Sections, Treated Geometrically by William Henry Besant

By William Henry Besant

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5. If P G, pg, the normals at the ends of a focal chord, intersect in O, the straight line through O parallel to P p bisects Gg. 6. Find the locus of the foci of all the conics of given eccentricity which pass through a fixed point P , and have the normal P G given in magnitude and position. 7. Having given a point P of a conic, the tangent at P , and the directrix, find the locus of the focus. 8. If P SQ be a focal chord, and X the foot of the directrix, XP and XQ are equally inclined to the axis.

Prove that SQ = 2SP , and that the ordinate of P is equal to the latus rectum. Also, if T is the point of intersection of the tangents at P and Q, and if R is the middle point of T Q, prove that the angle T SR is a right angle, and that ST = 2SR. 61. A straight line intersects a circle; prove that all the chords of the circle which are bisected by the straight line are tangents to a parabola. 62. If two tangents T P , T Q be drawn to a parabola, the perpendicular SE from the focus on their chord of contact passes through the middle point of their intercept on the tangent at the vertex.

A parabola is the curve traced out by a point which moves in such a manner that its distance from a given point is always equal to its distance from a given straight line. Tracing the Curve. 22. Let S be the focus, EX the directrix, and SX the perpendicular on EX. Then, bisecting SX in A, the point A is the vertex; and if, from any THE PARABOLA. 21 point E in the directrix, EAP , ESL be drawn, and from S the straight line SP meeting EA produced in P , and making the angle P SL equal to LSN , we obtain, as in Art.

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