Equation of hyperbola confocal with ellipse having same principal axes

AI Thread Summary
The discussion focuses on deriving the equation of a hyperbola that is confocal with a given ellipse, which has the equation (2x+3)²+(3y+2)²=8. The center of the ellipse is identified as (-3/2, -2/3), with calculated values for a², b², and eccentricity e. The participant expresses difficulty in finding the hyperbola's equation, noting that the principal axes may be unequal in length but coincident. They highlight that there is insufficient information to determine a unique hyperbola, suggesting that multiple hyperbolas could satisfy the given conditions. The conversation emphasizes the need for further clarification on verifying properties A and D without the hyperbola's equation.
Aurelius120
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Homework Statement
If the ellipse $$4x^2+9y^2+12x+12y+5=0$$ is confocal with a hyperbola having same principal axes then:
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The equation of ellipse reduces to :
$$(2x+3)^2+(3y+2)^2=8$$
$$\frac{(x+3/2)^2}{8/4}+\frac{(y+2/3)^2}{8/9}=1$$

Center of ellipse =##\left(\frac{-3}{2},\frac{-2}{3}\right)##

##b^2=a^2(1-e^2)=8/9## and ##a^2=8/4##
Therefore ##e=\frac{\sqrt{5}}{3}##

Distance between foci=##\frac{2\sqrt{10}}{3}##
Length of latus rectum of ellipse is 2 times the value of ##y## obtained on putting ##x=ae## and is equal to ##\frac{8\sqrt{2}}{9}##
Now comes the difficult part :
Finding the equation of hyperbola. I simply can't seem to do it.

For confocal hyperbola,
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{ and } ae=\frac{\sqrt{10}}{3}$$.
If principal axes have same length they become the same curve so I think principal axes are unequal in length but coincident.

I don't know what to do next ? I think there's insufficient information to get the equation of hyperbola.
 
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Also just to clarify more than one options are correct so all need to be checked
 
The question does not imply there is a unique hyperbola meeting those conditions. The question is whether every such hyperbola satisfies A and D.
 
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haruspex said:
The question does not imply there is a unique hyperbola meeting those conditions. The question is whether every such hyperbola satisfies A and D.
So without knowing the equation of hyperbol, how can A and D be verified?
 
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