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2. Zapiši enačbo hiperbole, če sta njeni asimptoti premici \( y= \pm 2 x \) in ima eno od temen v točki \( A(0,4) \). Hiperbolo tudi nariši ter zapiši koordinate gorišč.

Ask by Vega Byrd. in Slovenia
Mar 13,2025

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**Enačba hiperbole je** \[ \frac{y^2}{16} - \frac{x^2}{4} = 1 \] **in gorišča so** \[ (0,2\sqrt{5}) \quad \text{in} \quad (0,-2\sqrt{5}). \]

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Da bi zapisali enačbo hiperbole, moramo najprej ugotoviti njene osnovne podatke. Hiperbola ima v splošni obliki enačbo: \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \] kjer sta \((h, k)\) središče hiperbole, \(a\) je razdalja od središča do temena vzdolž y-osi, \(b\) pa razdalja od središča do asimptote. V naši situaciji so asimptote \(y = \pm 2x\), kar pomeni, da je odnos med \(a\) in \(b\) tak, da velja \(\frac{a}{b} = 2\). Temena hiperbole imata y-koordinato 4 in x-koordinato 0, kar pomeni, da je središče \(C(0,0)\) in da je \(a = 4\). Z uporabo razmerja načina asimptot dobimo \(b = 2\). Enačba hiperbole je torej: \[ \frac{y^2}{16} - \frac{x^2}{4} = 1 \] Hiperbolo lahko narišemo, pri čemer se njeni gorišči nahajajo na \( (0, 4) \) in \( (0, -4) \). Gorišči so zato točke \(F_1(0, 4)\) in \(F_2(0, -4)\). Upam, da ti je to v pomoč pri reševanju naloge!

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