Chapter 9: Q. 52 (page 772)

Show that the eccentricity satisfies the equation $B^{2} =$ $A^{2} (1 - e^{2})$ .

Short Answer

Expert verified

Hence proved.

Step by step solution

Step 1. Given information.

It is given $B^{2} = A^{2} (1 - e^{2})$ .

Step 2. Conclusion.

Now,

$B^{2} = A^{2} (1 - {(\frac{\sqrt{A^{2} - B^{2}}}{A})}^{2}) [since by substituting e = \frac{\sqrt{A^{2} - B^{2}}}{A}] = A^{2} (1 - \frac{{(\sqrt{A^{2} - B^{2}})}^{2}}{A^{2}}) = A^{2} (\frac{A^{2} - (A^{2} - B^{2})}{A^{2}}) = A^{2} (\frac{A^{2} - A^{2} + B^{2}}{A^{2}}) = A^{2} (\frac{B^{2}}{A^{2}}) B^{2} = B^{2}$

Hence proved.

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Show that the eccentricity satisfies the equation $B^{2} =$ $A^{2} (1 - e^{2})$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Conclusion.

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Show that the eccentricity satisfies the equation $B^{2} =$ $A^{2} (1 - e^{2})$ .