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Chapter 6: Problem 32

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$16 y^{2}-x^{2}+2 x+64 y+63=0$$

Short Answer

Expert verified
The hyperbola has center at (1, -2), vertices at (1 卤 鈭�67, -2), foci at (1 卤 鈭�83, -2), and equations of asymptotes as y = -2 卤 4/鈭�67 * (x - 1).

Step by step solution

01

Convert given equation into standard form

First, arrange the terms to form the equation in the general form of a hyperbola. Group the x's and y's, complete squares, and isolate the constants on the right side of the equation. \\ It's done like this: \\(16y^2 - x^2 + 2x + 64y + 63 = 0 \rightarrow -x^2 + 2x + 16y^2 + 64y = -63 \rightarrow 16(y^2 + 4y) - (x^2 - 2x) = -63 \\ After completing the square it becomes: \(16(y + 2)^2 - (x - 1)^2 - 67 = 0 \\ This simplifies to \\(x - 1)^2/67 - (y + 2)^2/16 = 1\)
02

Identify the center, vertices, and foci of the hyperbola

In the standard form of a hyperbola \\(\frac{{(x-h)^2}}{{a^2}} - \frac{{(y-k)^2}}{{b^2}} = 1\), the center is at the point (h, k). This gives the center as (1, -2). Also, a is the distance from the center to a vertex, so the vertices are at (1 卤 鈭�67, -2). The distance from the center to a focus is 鈭�(a^2 + b^2), which gives the foci at (1 卤 鈭�83, -2).
03

Find the equations of the asymptotes

The equations of the asymptotes of a hyperbola in standard form \\(\frac{{(x-h)^2}}{{a^2}} - \frac{{(y-k)^2}}{{b^2}} = 1\) are \\(y = k 卤 \frac{b}{a}(x - h)\). Substituting the values for h, k, a and b gives the equations of the asymptotes as \\(y = -2 卤 \frac{4}{鈭�67}(x - 1)\)
04

Sketch the hyperbola

Start by drawing the center of the hyperbola at the point (1, -2). Then sketch the asymptotes, and draw the hyperbola approaching these asymptotes. The vertices and foci can be plotted to aid with drawing.

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$3 x+5 y-2=0$$

Determine whether the statement is true or false. Justify your answer. If the asymptotes of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1,\) where \(a, b>0,\) intersect at right angles, then \(a=b\)

A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveyor. (c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

Consider the path of a projectile projected horizontally with a velocity of \(v\) feet per second at a height of \(s\) feet, where the model for the path is $$x^{2}=-\frac{v^{2}}{16}(y-s)$$ In this model (in which air resistance is disregarded), \(y\) is the height (in feet) of the projectile and \(x\) is the horizontal distance (in feet) the projectile travels. A ball is thrown from the top of a 100 -foot tower with a velocity of 28 feet per second. A. Find the equation of the parabolic path. B. How far does the ball travel horizontally before striking the ground?

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=2 \sin \theta$$
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