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Graph (x^2)/100-(y^2)/64=1

Problem

(x2)/100−(y2)/64=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=100 which means a=10 and b2=64 which means b=8

  3. Locate the vertices. For a horizontal hyperbola, the vertices are at (±a,0) which are (10,0) and (−10,0)

  4. Find the equations of the asymptotes. The asymptotes for a hyperbola centered at the origin are y=±b/a*x

  5. Substitute the values to get the specific asymptote equations.

y=±8/10*x

y=±4/5*x

  1. Calculate the foci using the relation c2=a2+b2

c2=100+64

c2=164

c=√(,164)=2√(,41)≈12.8

The foci are located at (±12.8,0)

  1. Sketch the graph by plotting the vertices, drawing the asymptotes through the origin, and drawing the two branches of the hyperbola opening to the left and right, approaching the asymptotes.

Final Answer

(x2)/100−(y2)/64=1* is a horizontal hyperbola with vertices *(±10,0)* and asymptotes *y=±4/5*x

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