# Calendar: 1879-1880 Page 596

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596 GENERAL LITERATURE AND SCIENCE ABC is triangle BD the third part of BC and CE the third part of CA find in what ratio AD and BE cut each other Draw tangents to parabola from an external point If PQ be the chord of contact of tangents from and PV be drawn parallel to the tangent at meeting the diameter at in prove that VT meets PQ in point of trisection Normals are drawn at the extremities of parallel chords of parabola prove that the locus of their intersection is straight line If from the foci Η of an ellipse SY HZ be drawn perpendicular to the tangent at show that Υ Ζ lie on the auxiliary circle Prove also that SZ HY bisect the normal at Define conjugate diameters Show that the general equation of pair of conjugate diameters of the curve ax1 -j- hxy by bm is mx -- xy y' Two of the normals to parabola from point Ο make equal angles with given straight line Prove that the locus of Ο is parabola Prove that the chord of an ellipse which subtends right angle at its pole always touches confocal ellipse PQ is any diameter of an ellipse Ε is any fixed point and the line from Ε to the centre of the ellipse cuts it in Show that the locus of the intersections of EP and AQ is similar ellipse 10 If rectangular hyperbola circumscribe triangle its centre lies on the nine point circle 11 system of similar curves is described about triangle prove that the locus of their centres is curve of the fourth degree touching each side of the triangle at its middle point 12 Prove that λα2 μβ2 νγ1 represents curve and find the pole of the line pa ςβ ry with respect to it

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