Questions: Graph the hyperbola. [ (y-3)^2/25-(x-4)^2/16=1 ]

Transcript text: Graph the hyperbola. \[ \frac{(y-3)^{2}}{25}-\frac{(x-4)^{2}}{16}=1 \]

Solution

Solution Steps

Step 1: Identify the Hyperbola Equation

The given equation of the hyperbola is: \[ \frac{(y-3)^{2}}{25}-\frac{(x-4)^{2}}{16}=1 \]

Step 2: Determine the Center of the Hyperbola

The center of the hyperbola is at the point $(h, k)$, where $h = 4$ and $k = 3$.

Step 3: Identify the Transverse and Conjugate Axes

The transverse axis is vertical because the $y$-term is positive. The length of the semi-transverse axis is $a = \sqrt{25} = 5$, and the length of the semi-conjugate axis is $b = \sqrt{16} = 4$.

Final Answer

The hyperbola is centered at $(4, 3)$ with a vertical transverse axis of length 10 and a horizontal conjugate axis of length 8.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 18, "ymin": -10, "ymax": 16}, "commands": ["(y-3)2/25 - (x-4)2/16 = 1"], "latex_expressions": ["$\\frac{(y-3)^{2}}{25}-\\frac{(x-4)^{2}}{16}=1$"]}

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