<prompt> Acha T_1, T_2, T_3 ziwe pointi kwenye kifananisho, na t_1, t_2, t_3 dondoo za kifananisho katika pointi hizi. Kokotoa uwiano wa eneo la pembetatu T_1 T_2 T_3 kwa eneo la pembetatu inayobainishwa na tanjiti.  <response>: Chagua mfumo wa Cartesian wa kuratibu ili kwamba mlingano wa parabola ni y24 p x. Viwianishi vya pointi tatu ni T_ileft4 p alpha_i2, 4 p alpha_ikulia, kwa alpha_i iliyochaguliwa ipasavyo, i 1,2,3. Kumbuka kwamba mlinganyo wa tanjiti kwa kiambatanisho katika hatua leftx_0, y_0kulia ni y y_02 pleftxx_0 kulia . Katika hali yetu tanjenti tatu zinatolewa na 2 alpha_i yx4 p alpha_i2, quad i1,2,3 . Kama P_i j ni makutano ya t_i na t_j, basi viwianishi vyake ni left4 p alpha_i alpha_j, 2 p. kushotoalpha_ialpha_jkuliakulia. Eneo la pembetatu T_1 T_2 T_3 limetolewa na kibainishi cha Vandermonde: pm frac12leftbeginarraylll 4 p. alpha_12 4 p alpha_1 1 4 p alpha_22 4 p alpha_2 1 4 p alpha_3 2 4 p alpha_3 1 mwishosafukuliapm 8 p2kushotoanzaarraylll alpha_1 2 alpha_1 1 alpha_22 alpha_2 1 alpha_32 alpha_3 1 mwisho arrayright8 p2leftleftalpha_1-alpha_2rightleftalpha_1-alpha_3right kushotoalpha_2-alpha_3kuliakulia . Eneo la pembetatu P_12 P_23 P_31 imetolewa na beginaligned pm frac12leftbeginarray lll4 p alpha_1 alpha_2 2 pkushotoalpha_1alpha_2kulia 1 4 p alpha_2 alpha_3 2 pkushotoalpha_2alpha_3kulia 1 4 p alpha_3 alpha_1 2 pleftalpha_3 alpha_1kulia na 1mwishosafukulia pm 4 p2leftanzaarrayllalpha_1 alpha_2leftalpha_1alpha_2kulia 1 alpha_2 alpha_3kushotoalpha_2alpha_3kulia na 1 alpha_3 alpha_1kushotoalpha_ 3alpha_1kulia 1mwishosafukuliapm 4 p2leftanzaarrayrrleftalpha_1 -alpha_3kulia alpha_2kushotoalpha_1-alpha_3kulia 0 kushotoalpha_2-alpha_1 kulia alpha_3kushotoalpha_2-alpha_1kulia na 0 alpha_3 alpha_1kushotoalpha_3alpha_ 1kulia na 1mwishosafukulia 4 p2kushotokushotoalpha_1-alpha_3kuliakushotoalpha_1-alpha_2kulialeft alpha_2-alpha_3kuliakulia maandishi . endaligned Tunahitimisha kuwa uwiano wa maeneo mawili ni 2, bila kujali eneo la pointi tatu au umbo la parabola. 