4.1 Geometry 215604.Consider a circle of diameterABand centerO, and the tangenttatB. A variable
tangent to the circle with contact pointMintersectstatP. Find the locus of the
pointQwhere the lineOMintersects the parallel throughPto the lineAB.
605.On the axis of a parabola consider two fixed points at equal distance from the focus.
Prove that the difference of the squares of the distances from these points to an
arbitrary tangent to the parabola is constant.
606.With the chordPQof a hyperbola as diagonal, construct a parallelogram whose
sides are parallel to the asymptotes. Prove that the other diagonal of the parallelo-
gram passes through the center of the hyperbola.
607.A straight line cuts the asymptotes of a hyperbola in pointsAandBand the hyperbola
itself inPandQ. Prove thatAP=BQ.
608.Consider the parabolay^2 = 4 px. Find the locus of the points such that the tangents
to the parabola from those points make a constant angleφ.
609.LetT 1 ,T 2 ,T 3 be points on a parabola, andt 1 ,t 2 ,t 3 the tangents to the parabola at
these points. Compute the ratio of the area of triangleT 1 T 2 T 3 to the area of the
triangle determined by the tangents.
610.Three pointsA, B, Care considered on a parabola. The tangents to the parabola
at these points form a triangleMNP(NPbeing tangent atA,PMatB, andMN
atC). The parallel throughBto the symmetry axis of the parabola intersectsAC
atL.
(a) Show thatLMN Pis a parallelogram.
(b) Show that the circumcircle of triangleMNPpasses through the focusFof the
parabola.
(c) Assuming thatLis also on this circle, prove thatNis on the directrix of the
parabola.
(d) Find the locus of the pointsLifACvaries in such a way that it passes through
Fand is perpendicular toBF.
- Find all regular polygons that can be inscribed in an ellipse with unequal semiaxes.
612.We are given the parabolay^2 = 2 pxwith focusF. For an integern≥3 consider
a regular polygonA 1 A 2 ...Anwhose center isFand such that none of its vertices
is on thex-axis. The half-lines|FA 1 ,|FA 2 ,...,|FAnintersect the parabola atB 1 ,
B 2 ,...,Bn. Prove that
FB 1 +FB 2 +···+FBn≥np.613.Acevianof a triangle is a line segment that joins a vertex to the line containing the
opposite side. Anequicevian pointof a triangleABCis a pointP(not necessarily