Shifting Conic Sections

You may wish to review Section 1.2 before solving Exercises 39-56.

Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.

x²/4 − y²/5 = 1, right 2, up 2

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Identify the original hyperbola equation: \(\frac{x^{2}}{4} - \frac{y^{2}}{5} = 1\). This is a hyperbola centered at the origin \((0,0)\) with the transverse axis along the x-axis.

Apply the given shifts: right 2 units and up 2 units. This means replacing \(x\) by \((x - 2)\) and \(y\) by \((y - 2)\) in the equation to shift the center to \((2, 2)\).

Write the new equation of the hyperbola after shifting: \(\frac{(x - 2)^{2}}{4} - \frac{(y - 2)^{2}}{5} = 1\).

Determine the new center from the shifts: the center is now at \((2, 2)\).

Find the new vertices, foci, and asymptotes by using the standard formulas for hyperbolas centered at \((h, k)\): - Vertices: \((h \pm a, k)\) where \(a^{2} = 4\). - Foci: \((h \pm c, k)\) where \(c^{2} = a^{2} + b^{2} = 4 + 5 = 9\). - Asymptotes: lines through the center with slopes \(\pm \frac{b}{a} = \pm \frac{\sqrt{5}}{2}\), so equations are \(y - k = \pm \frac{\sqrt{5}}{2}(x - h)\).

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주요 개념

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Standard Form of a Hyperbola

A hyperbola's equation in standard form is typically written as (x - h)²/a² - (y - k)²/b² = 1 or its vertical counterpart. Here, (h, k) is the center, and a and b determine the distances to vertices and asymptotes. Understanding this form helps identify key features like center, vertices, foci, and asymptotes.

Translation (Shifting) of Conic Sections

Shifting a conic involves replacing x with (x - h) and y with (y - k) to move the graph h units right/left and k units up/down. This changes the center of the conic without altering its shape. Applying translations correctly updates the equation and the coordinates of key points.

Finding Foci and Asymptotes of a Hyperbola

The foci of a hyperbola lie along the transverse axis, found using c² = a² + b², where c is the focal distance from the center. Asymptotes are lines that the hyperbola approaches, with slopes ±b/a (horizontal transverse axis) or ±a/b (vertical). Calculating these requires knowing a, b, and the center.