Graph (x^2)/64-(y^2)/36=1

Problem

(x2)/64−(y2)/36=1

Solution

1. Identify the type of conic section. Since the equation is in the form (x2)/(a2)−(y2)/(b2)=1 it represents a horizontal hyperbola centered at the origin (0,0)

2. Determine the values of a and b We have a2=64 so a=8 We have b2=36 so b=6

3. Locate the vertices. For a horizontal hyperbola, the vertices are at (±a,0) which are (8,0) and (−8,0)

4. Find the foci. Use the relation c2=a2+b2

c2=64+36

c2=100

c=10

The foci are at (±c,0) which are (10,0) and (−10,0)

5. Identify the asymptotes. The equations for the asymptotes are y=±b/a*x

y=±6/8*x

y=±3/4*x

6. Sketch the graph. Draw a central rectangle extending from −8 to 8 on the x-axis and −6 to 6 on the y-axis. Draw the diagonal asymptotes through the corners of this rectangle, then draw the two branches of the hyperbola opening left and right from the vertices.

Final Answer

(x2)/64−(y2)/36=1* is a horizontal hyperbola with vertices *(±8,0)* and asymptotes *y=±3/4*x

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