Problem 48 Graph the hyperbola whose equati... [FREE SOLUTION]

Chapter 11: Problem 48

Graph the hyperbola whose equation is $$ 25 x^{2}-16 y^{2}-100 x-96 y-444=0 $$ Where are the foci located? What are the equations of the asymptotes?

Short Answer

Expert verified

The foci of the hyperbola are located at (5.2,-3) and (-1.2,-3). The equations of the asymptotes are $y = -3 + 1.25(x - 2)$ and $y = -3 - 1.25(x - 2)$

Step by step solution

Rewriting the Equation to Standard Form

The general form of a hyperbola is given as $Ax^2 + By^2 + Dx + Ey + F = 0$. And the standard form is $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ where center is (h,k), a is the length of the transverse axis and b is the length of the conjugate axis. Rewriting the given equation to resemble the standard form, you would get $25(x-2)^2 - 16(y+3)^2 = 100$.

Identify the Parameters

From the standard form, the center of the hyperbola can be found as (h,k) which is (2,-3) for this equation. You can also identify a and b, the lengths of the axes, where $a^2 = \frac{100}{25} = 4$ and $b^2 = \frac{100}{16} = 6.25$. This means a = 2 and b = 2.5, hence the length of transverse axis is 2 and the conjugate axis is 2.5 respectively.

Determining the Foci

The distance from the center to each focus is given by $c = \sqrt{a^2 + b^2}$ in the case of hyperbola. Using our values for a and b, we find $c = \sqrt{4 + 6.25} = \sqrt{10.25} = 3.2$. Hence, the foci are at $(h+c, k)$ and $(h-c, k)$ which will be $(2+3.2, -3)$ and $(2-3.2, -3)$

Determine the Equation of Asymptotes

The equations of the asymptotes for a hyperbola are always in the form $y = k \pm \frac{b}{a}(x - h)$. Substituting for h, k, a and b, we get $y = -3 \pm \frac{2.5}{2}(x - 2)$. Hence the equations will be $y = -3 + 1.25(x - 2)$ and $y = -3 - 1.25(x - 2)$.

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91影视

Graph the hyperbola whose equation is $$ 25 x^{2}-16 y^{2}-100 x-96 y-444=0 $$ Where are the foci located? What are the equations of the asymptotes?

Short Answer

Step by step solution

Rewriting the Equation to Standard Form

Identify the Parameters

Determining the Foci

Determine the Equation of Asymptotes

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