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Chapter 10: Q.37 (page 699)

Find an equation of the hyperbola whose foci are the vertices of the ellipse 4x2+9y2=36and whose vertices are the foci of this ellipse.

Short Answer

Expert verified

The equation of the hyperbola isx25-y24=1

Step by step solution

01

Step 1. Given Information

Given to determine the equation of the hyperbola whose foci are the vertices of the ellipse 4x2+9y2=36and whose vertices are the foci of this ellipse.

02

Step 2. Analyzing the ellipse

Rewriting the given equation of ellipse:

4x236+9y236=3636x29+y24=1

Here a=3,b=2

Hence the vertices of the parabola are -3,0,3,0

The foci of this ellipse lie at -c,0,c,0where c2=a2-b2.

Plugging the values:

localid="1648614483350" c2=a2-b2c2=9-4c=卤5

Hence the foci of the ellipse are-5,0,5,0

03

Step 3. Equation of hyperbola

So the vertices of the hyperbola are -5,0,5,0and the foci are -3,0,3,0.

Here, role="math" localid="1648122490904" a=5,c=3

For a hyperbola,

c2=a2+b29=5+b2b=2

Hence the equation of the hyperbola is given by:

x2a2-y2b2=1x25-y24=1

04

Step 4. Conclusion

The equation of the hyperbola isx25-y24=1

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