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The Smith Geometry Problems

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The Smith Geometry Problems The Smith Geometry Problems By Dr. Paul Mills Ph.D Kenilworth,England,24th March 2003 The Smith Geometry problems are 100 advanced problems in co-ordinate geometry collected and published by Charles Smith (former Master of Sidney Sussex College, Cambridge University) in 1882. Solutions are invited using the dynamic geometry program Euklides available at www.euklides.hu. Algebraic solutions are also wel- come (*.pdf file format only). Upload your dynamic geometry solutions (Euklides files *.euk only) to the Files area of the Dynamic Geometry Yahoo Group at the internet address http://groups.yahoo.com/group/dynamicgeometry. Keywords: The Smith Geometry problems 1. The Smith Geometry Problems Q.1 If Q and P be any two points, and C the centre of a conic; shew that the perpendiculars from Q and C on the polar of P with respect to the conic, are to one another in the same ratio as the perpendiculars from P and C on the polar of Q. Q.2 Two tangents drawn to a conic from any point are in the same ratio as the corresponding normals Q.3 (i) All chords of a conic which subtend a right angle at a fixed point O on the conic, cut the normal at O in a fixed point. (ii) If any two chords OP,OQ of a conic make equal angles with the tangent at O, the line PQ will cut that tangent in a fixed point. Q.4 P OQ is one of a system of parallel chords of an ellipse, and O is the point on it such that PO2 + OQ2 is contant; shew that, for different positions of the chord, the locus of O is a concentric conic. Q.5 If O be any fixed point and OPP’ any chord cutting a conic in P,P’, and on this line a point D be taken such that 1 1 1 = + (1.1) OD2 OP 2 OP 02 the locus of D will be a conic whose centre is O. Q.6 If OPP’QQ’ is one of a system of parallel straight lines cutting one given conic in P, P’ and another in Q,Q’ and O is such that the ratio of the rectangles OP.OP’ and TEX Paper 2 The Smith Geometry Problems OP.OQ’ is constant; shew that the locus of O is a conic through the intercessions of the original conics. Q.7 POP’, QOQ’ are any two chords of a conic at right angles to one another through a fixed point O; shew that 1 1 + (1.2) P O.OP 0 QO.OQ0 is constant. Q.8 If a point be taken on the axis-major of an ellipse, whose abscissa is equal to ra2 − b2 a (1.3) a2 + b2 prove that the sum of the squares of the reciprocals of the segments of any chord passing through that point is constant. Q.9 If PP’ be any one of a system of parallel chords of a rectangular hyperbola, and A,A’ be the extremities of the perpendicular diameter; PA and P’A’ will meet on a fixed circle. Shew also that the words rectangular hyperbola, and circle, can be interchanged. Q.10 If PSP’ be any focal chord of a parabola and PM, P’M’ be perpendiculars on a fixed straight line, then will PM P 0M 0 + (1.4) PS0 P 0S be constant. Q.11 Chords of a circle are drawn through a fixed point and circles are described on them as diameters; prove that the polar of the point with regard to any one of these circles touches a fixed parabola. Q.12 From a fixed point on a conic chords are drawn making equal intercepts, mea- sured from the centre, on a fixed diameter; find the locus of the point of intersection of the tangents at their other extremities. Q.13 If (x’,y’) and (x”,y”) be the co-ordinates of the extremities of any focal chord of an ellipse, and x, y be the co-ordinates of the middle point of the chord; shew that y’,y” will vary as x. What does this become for a parabola? Q.14 S,H are two fixed points on the axis of an ellipse equidistant from the centre C; PSQ, PHQ’ are chords through them, and the ordinate MQ is produced to R so that MR may be equal to the abscissa of Q’; shew that the locus of R is a rectangular hyperbola. Q.15 S,H are two fixed points on the axis of an ellipse equidistant from the centre, and PSQ, PHQ’ are two chords of the ellipse; shew that the tangent at P and the line QQ’ make angles with the axis whose tangents are in a constant ratio. The Smith Geometry Problems 3 Q.16 Two parallel chords of an ellipse, drawn through the foci, intersect the curve in points P,P’ on the same side of the major axis, and the line through P, P’ intersects the semi-axes CA,CB in U,V respectively: prove that AC4 BC4 + (1.5) UC2 VC2 is invariable. Q.17 From an external point two tangents are drawn to an ellipse; shew that if the four points where the tangents cut the axes lie on a circle, the point from which the tangents are drawn will lie on a fixed rectangular hyperbola. Q.18 Prove that the locus of the intersection of tangents to an ellipse which make equal angles with the major and minor axes respectively, but which are not at right angles, is a rectangular hyperbola whose vertices are the foci of the ellipse. Q.19 If a pair of tangents to a conic meet a fixed diameter in two points such that the sum of their distances from the centre is constant; shew that the locus of the point of intersection is a conic. Shew also that the locus of the point of intersection is a conic if the product, or if the sum of the reciprocals be constant. Q.20 Through O, the middle point of a chord AB of an ellipse, is drawn any chord POQ. The tangents at P and Q meet AB in S and T respectively. Prove that AS=BT. Q.21 Pairs of tangents are drawn to the conic ax2 + by2 = 1 (1.6) so as to be always parallel to conjugate diameters of the conic ax2 + 2hxy + by2 = 1 (1.7) Shew that the locus of their intersection is a b ax2 + by2 + 2hxy = + (1.8) α β Q.22 PT,PT’ are two tangents to an ellipse which meet the tangent at a fixed point Q in T,T’; find the locus of P (i) when the sum of the squares of QT and QT’ is constant, and (ii) when the rectangle QT.QT’ is constant. Q.23 O is a fixed point on the tangent at the vertex A of a conic, and P,P’ are points on that tangent equally distant from O; shew that the locus of the point of intersection of the other tangents from P and P’ is a straight line. Q.24 If from any point of the circle circumscribing a given square tangents be drawn to the circle inscribed in the same square, these tangents will meet the diagonals of the square in four points lying on a rectangular hyperbola. 4 The Smith Geometry Problems Q.25 Find the locus of the point of intersection of two tangents to a conic which intercept a constant length on a fixed straight line. Q.26 Two tangents to a conic meet a fixed straight line MN in P,Q: if P,Q be such that PQ subtends a right angle at a fixed point O, prove that the locus of the point of intersection of the tangents will be another conic. Q.27 The extremities of the diameter of a circle are joined to any point, and from that point two tangents are drawn to the circle; shew that intercept on the perpendicular diameter between one line and one tangent is equal to that between the other line and the other tangent. Q.28 Triangles are described about an ellipse on a given base which touches the ellipse at P; if the base angles be equidistant from the centre, prove that the locus of their vertices is the normal at the other end of the diameter through P. Q.29 A parabola slides between rectangular axes; find the curve traced out by any point in its axis; and hence shew that the focus and vertex will describe curves of which the equations are x2y2 = a2(x2 + y2) (1.9) x2y2(x2 + y2 + 3a2) = a2 (1.10) 4a being the latus rectum of the parabola. Q.30 If the axes of co-ordinates be inclined to one another at an angle α, and an ellipse slide between them, shew that the equation of the locus of the centre is sin2α(x2 + y2 − p2)2 − 4cos2α(x2y2sin2α − q4) = 0 (1.11) where p2 and q2 denote respectively the sum of the squares and the product of the semi-axes of the ellipse. Q.31 If OP, OQ are two tangents to an ellipse, and CP’, CQ’ the parallel semi- diameters, shew that OP.OQ + CP 0.CQ0 = OS.OH (1.12) S,H being the foci. Q.32 Through two fixed points P,Q straight lines APB, CQD are drawn at right angles to one another, to meet one given straight line in A,C and another given straight line perpendicular to the former in B,D; find the locus of the point of intersectionof AD, BC; and shew that, if the line joining P and Q subtend a right angle at the point of intersection of the given lines, the locus will be a rectangular hyperbola.
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