1. Se da familia de parabole :
![\[y=(m+2)x^2-3(m-1)x+m ,\;\;m\in R\{-2}\]](https://anidescoala.ro/wp-content/ql-cache/quicklatex.com-a56f91dbb06e2af094f8761925cc680a_l3.png "Rendered by QuickLaTeX.com")
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a). Sa se determine m astfel incat varful parabolelor sa se gaseasca in multimea D={(x,y)|x<0,y<0,y<x}.
b). Pentru ce valori ale lui m parabolele date sunt tangente dreptei de ecuatie y=2x-1.
c). Sa se arate ca parabolele definite mai sus trec prin doua puncte fixe, ale caror coordonate se cer.
Rezolvare:
![\[\begin{array}{l} a). V\left( { - \frac{b}{{2a}}, - \frac{\Delta }{{4a}}} \right) \\ x_V = - \frac{b}{{2a}} = \frac{{3(m - 1)}}{{2(m + 2)}}\quad ;y_v = - \frac{\Delta }{{4a}} = - \frac{{5m^2 - 26m + 9}}{{4(m + 2)}} \\ \Rightarrow \left\{ \begin{array}{l} x_v < 0 \\ y_v < 0 \\ y_v < x{}_v \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \frac{{3(m - 1)}}{{2(m + 2)}} < 0 \\ - \frac{{5m^2 - 26m + 9}}{{4(m + 2)}} < 0 \\ - \frac{{5m^2 - 26m + 9}}{{4(m + 2)}} < \frac{{3(m - 1)}}{{2(m + 2)}} \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m \in ( - 2,1) \\ m \in \left( { - 2,\frac{{13 - 2\sqrt {31} }}{5}} \right) \cup \left( {\frac{{13 + 2\sqrt {31} }}{5},\infty } \right) \\ m \in \left( { - 2, - \frac{{\sqrt {85} }}{5} + 2} \right) \cup \left( {\frac{{\sqrt {85} }}{5} + 2} \right) \\ \end{array} \right. \\ \Prin \; intersectie \Rightarrow m \in \left( { - 2,\frac{{\sqrt {85} }}{5} + 2} \right) \\ \end{array}\]](https://anidescoala.ro/wp-content/ql-cache/quicklatex.com-b62da0257b1f17382d7bf0190c1d976e_l3.png "Rendered by QuickLaTeX.com")
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b).parabole\;\tan gente\;dreptei\;d]
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Emergency stop.
![\[\begin{array}{l} c). y = mx^2 + 2x^2 - 3mx + 3x + m = \left( {x^2 - 3x + 1} \right)m + 2x^2 + 3x \\ \Rightarrow \left\{ \begin{array}{l} x^2 - 3x + 1 = 0 \\ 2x^2 + 3x = y \\ \end{array} \right. \Rightarrow A\left( {\frac{{3 - \sqrt 5 }}{2},\frac{{23 - 9\sqrt 5 }}{2}} \right)\quad si\;B = \left( {\frac{{3 + \sqrt 5 }}{2},\frac{{23 + 9\sqrt 5 }}{2}} \right) \\ \end{array} \]](https://anidescoala.ro/wp-content/ql-cache/quicklatex.com-558b6d4d137e9dd00e843f9295be69a0_l3.png "Rendered by QuickLaTeX.com")
Este bine??
Cred ca-i bine. N-am stat sa verific calculele dar daca ai aplicat formulele corect inseamna ca e foarte bine.