Hyperbolas and Eccentricity

In Exercises 17-2...

Question

Hyperbolas and Eccentricity

In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.

y² − x² = 4

Accepted Answer

For the hyperbole x2 minus y2 equals 9, find the eccentricity. Then graph the hyperbole and mark the foci in directors on your graph. We're also given a graph to plot our equation on. Now, we first need to write this in standard form. We have x2 minus y2 equals 9, so let's divide both sides by 9. This gives us X2 divided by 9 minus Y2 divided by 9 equals 1. We now have A2 equals 9 and B2 equals 9, meaning A equals 3 and B equals 3. Now, to find eccentricity, we need to find C first, which is the distance to the phosite. With hyperbole, we know that C is C2 equals A2 + B2. We have C2 equals 9 + 9, or C2 equals 18, meaning C is 32 of 2 when we take the square root of both sides. This means the foci are located at plus or minus +32 20. Now we can use this to find eccentricity. Eccentricity E is given by C. Divided by a. This tells us how stretched the hyperbola is. We have 3 squares of 2 divided by 3, which this simplifies to give us the square of 2. Now, we can use this to find our directracity equations. This is given as X equals plus or minus A divided by E, or 3 divided by the square root of 2. Now, we can graph our equation. We first know that the center is at 00 because this is not shifted. So we'll plot a point at 00. Next, we know that A is 3. So, what we can do there. I add 3 to the X coordinate of our vertex. That gives us -30 and 30 for our vertices. And our foci are at plus or minus +23. Or plus or minus 3 squares of 2 rather. These are just barely. Past our vertices. So we'll put a point. At roughly the square of 18. This is a number that's between 4 and 5, so we'll put a point there. On both sides. And now we can plot our directories. Which are at 3 divided by the square of 2 and -3 divided by the square root of 2. This is a number that is just above 2, so we'll draw a dashed line. Just to the right of 2 and to the left. Of -2. Now we can draw a sketch of our graph. We'll start from the vertex, and kind of just draw a hyperbola curving around. Our foa We go up to the right, down to the right. Up to the left. What should be mirrored. And down to the left. This is the equation of our graph, approximately. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Hyperbolas and EccentricityIn Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.y² − x² = 4

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Hyperbolas and Eccentricity

In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.

y² − x² = 4