GEOMETRIA 373 Disegna la parabola di equazione 1 3 13 y = __x2 __x + ___ e scrivi le equazioni delle rette 4 2 4 a essa tangenti rispettivamente nel suo punto P di ascissa 1 e nel suo simmetrico rispetto all asse della parabola, indicato con Q. Calcola l area del triangolo PVQ essendo V il vertice della parabola. [y = x + 3; y = x 3; 2] 375 Scrivi l equazione della parabola (con asse paral- lelo all asse delle ordinate) tangente alla retta di equazione y = x + 6 e con il vertice sull asse delle ascisse nel punto di ascissa 8. _1_ 2 [y = 8 x 2x + 8] 376 Sull asse della parabola di equazione 374 Scrivi l equazione della parabola (con asse paral- lelo all asse delle ordinate) di vertice O e tangente 1 _1_ 2 alla retta di equazione y = x __. [y = 2 x ] 2 1 y = __x2 + 2x + 3 considera il punto A di ordina2 13 ta ___ e traccia per esso le due rette tangenti alla 2 parabola. Indicati con B e C i loro punti di tangenza, verifica che il triangolo ABC è equilatero e cal__ [2 3 ] cola la lunghezza del suo lato. PRACTICE WITH CLIL How to make a parabola by folding a piece of paper Get a rectangle shaped paper and draw a dot F on it. The dot is the parabola focus and the side of the paper is its directrix. Then, fold the longest side of the paper so that it lines up with F. On the crease, you will certainly find a point of the parabola having as focus F and as directrix the side of the paper: the crease is the tangent of the parabola. Now crease the paper many times, at least 20 times in different positions, so that the folded side crosses the dot. The creases are the tangents of the parabola; they will approximately determine the paF F F rabola. Now you can determine, for each tangent, the point that belongs to the parabola. To do this, indicate with d the straight edge of the paper (directrix), with H the point on the side of the paper that coincides with F in a crease: points F and H are symmetrical with respect to the straight line s identified by the fold. s P F d F d H Now draw the perpendicular line to the side passing through H and indicate with P the point of intersection of the perpendicular with the crease. (You can also obtain the perpendicular line by making a fold perpendicular to the edge, which passes through H: this line intersects the crease in P). P is a point in the parabola. In fact: P belongs to the fold, which is axis of the segment HF and therefore equidistant from the extremes: d(P, F) = d(P, H); the distance of P from the side of the paper is the length of PH, that is d(P, d) = d(P, H); for the transitive relation: d(P, F) = d(P, d). Therefore, P is a point in the parabola, which is, as you know, the locus of points that are equidistant from a given point (called focus) and a line (called directrix). s (folding) t P F H r Exercise Draw the perpendicular line to the side of the paper, which is the parabola directrix crossing the focus: the parabola s axis of symmetry. Spot the vertex (the midpoint between the focus and the directrix on the axis) and draw a line parallel to the directrix. By indicating the two lines as the Cartesian axes and measuring the distance of the focus from the vertex with the ruler, determine the parabola s equation plotted out by the folded paper. You can repeat the exercise by changing the position of F on the sheet of paper. 462