Chapter 10: Q.37 (page 699)

Find an equation of the hyperbola whose foci are the vertices of the ellipse $4 x^{2} + 9 y^{2} = 36$ and whose vertices are the foci of this ellipse.

Short Answer

Expert verified

The equation of the hyperbola is $\frac{x^{2}}{5} - \frac{y^{2}}{4} = 1$

Step by step solution

Step 1. Given Information

Given to determine the equation of the hyperbola whose foci are the vertices of the ellipse $4 x^{2} + 9 y^{2} = 36$ and whose vertices are the foci of this ellipse.

Step 2. Analyzing the ellipse

Rewriting the given equation of ellipse:

$\frac{4 x^{2}}{36} + \frac{9 y^{2}}{36} = \frac{36}{36} \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$

Here $a = 3, b = 2$

Hence the vertices of the parabola are $(- 3, 0), (3, 0)$

The foci of this ellipse lie at $(- c, 0), (c, 0)$ where $c^{2} = a^{2} - b^{2}$ .

Plugging the values:

localid="1648614483350" $c^{2} = a^{2} - b^{2} c^{2} = 9 - 4 c = \pm \sqrt{5}$

Hence the foci of the ellipse are $(- \sqrt{5}, 0), (\sqrt{5}, 0)$

Step 3. Equation of hyperbola

So the vertices of the hyperbola are $(- \sqrt{5}, 0), (\sqrt{5}, 0)$ and the foci are $(- 3, 0), (3, 0)$ .

Here, role="math" localid="1648122490904" $a = \sqrt{5}, c = 3$

For a hyperbola,

$c^{2} = a^{2} + b^{2} 9 = 5 + b^{2} b = 2$

Hence the equation of the hyperbola is given by:

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \frac{x^{2}}{5} - \frac{y^{2}}{4} = 1$

Step 4. Conclusion

The equation of the hyperbola is $\frac{x^{2}}{5} - \frac{y^{2}}{4} = 1$

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Find an equation of the hyperbola whose foci are the vertices of the ellipse $4 x^{2} + 9 y^{2} = 36$ and whose vertices are the foci of this ellipse.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Analyzing the ellipse

Step 3. Equation of hyperbola

Step 4. Conclusion

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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

Step by step solution

Step 1. Given Information

Step 2. Analyzing the ellipse

Step 3. Equation of hyperbola

Step 4. Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Geometry

Mechanics Maths

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find an equation of the hyperbola whose foci are the vertices of the ellipse $4 x^{2} + 9 y^{2} = 36$ and whose vertices are the foci of this ellipse.