Question

The point ?(√6 cos ? , √3 sin ?) is on an ellipse.

a) Write down the equation of this ellipse in Cartesian form and find its foci.

b) A hyperbola has the same foci as this ellipse and one of the branches cuts the ?-axis at 1. What is the equation of the hyperbola?

Answer 1

a) the given point is

we hve to find the cartesian equation of the ellipse

for that let

  

we have to eliminate from equations (1) and (2) to get the cartesian equation

from (1) we get

  

from (2) we get

we know that      

then

  

, which is the required cartesian equation of the ellipse

now to find focii of the ellipse

[ if the given ellipse is with a>b its focii is given by    ]

here the equation of the ellipse is

  

from the equation we get

then   

then the focii are  given by

b) here we have to find the equation of hyperbola with focii

since focii lie on x axis the x axis form the transverse axis for the hyperbola

also given that one of the branch of the hyperbola cuts the x axis at 1

so it forms one of the vertex (1,0)

that is vertex ,  (a,0) = (1,0)

we know that focii of a hyperbola is given by where e is the eccentricity

here given focii =

that is

then

we know that for a hyperbola

=

since x axis the transverse axis , the equation of the hyperbola is

  

here substitute a2 =1 and b2=2

   , the required hyperbola