Confocal ellipse and hyperbola

[Ananya0107]

If a hyperbola passes through the focii of the ellipse x^2/25 +y^2/16 =1 and its transverse and conjugate axes coincide respectively with major and minor axes of the ellipse, and if the product of eccentricities of hyperbola and ellipse is 1, find the equation and focus of the hyperbola

 

[AdityaDev]

There is a very important property regarding confocal ellipse and hyperbola.
"When ellipse and hyperbola are confocal, then they are orthogonal curves"

 

[AdityaDev]

But I don't think you need that property here. I was thinking about finding the differential equation for the ellipse and then substituting -dx/dy for dy/dx and then finding the curve equation for hyperbola by integrating. You can try it.

 

[AdityaDev]

A second method is to find e for ellipse, then find focus, then the hyperbola equation by using the information of product of eccentricities.

 

[Ananya0107]

Actua

But I don't think you need that property here. I was thinking about finding the differential equation for the ellipse and then substituting -dx/dy for dy/dx and then finding the curve equation for hyperbola by integrating. You can try it.

actually I was thinking too much ... Eccentricity of the hyperbola = 5/3 from the question , and it passes through (±3, 0) , its equation therefore is x^2/9 - y^2/b^2 =1 where 1+ b^2/9 = 25/9 therefore equation of hyperbola is x^2/9 - y^2/9 = 1 and its focii are ±5,0

 

[AdityaDev]

Coordinate geometry is one topic which tests how much properly you can use the given information. But there are very difficult questions in this topic.

 

[Ananya0107]

True..