Find Eccentricity of Conic Passing Through Origin with Focii (5,12) and (24,7)

[hellboydvd]

If (5,12) and (24,7) are the focii of a conic passing through the origin, then find the eccentricity of the conic

Attempt:

Found the centre as (h,k), midpoint of the given points. (x-h)^2/a^2+(y-k)^2/b^2=1 i put x=0 and y=0 as it passes through the origin. from the equation e^2=1-(b^2/a^2) i got a relation between a and b. distance between the two given points equals 2a*e thereby giving the second relation, manipulating both the equations, i got a quadratic in e which i solved but it became tedious and finally i got the wrong answer.

Help would be appreciated!

 

[tiny-tim]

Welcome to PF!

Hi hellboydvd ! Welcome to PF!

Hint: what is the sum of the distances of the foci (not focii!) from the origin, in terms of a b and e?

 

[HallsofIvy]

But notice that the form (x-h)^2/a^2+(y-k)^2/b^2=1 is only valid for an ellipse with axes parallel to the coordinate axes. Fortunately, you don't need the equation of the ellipse to do this problem.