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

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SUMMARY

The discussion focuses on finding the eccentricity of a conic section that passes through the origin with foci at (5,12) and (24,7). The midpoint of the foci is calculated to determine the center, and the relationship between the semi-major axis (a), semi-minor axis (b), and eccentricity (e) is established using the equation e² = 1 - (b²/a²). A hint is provided that emphasizes the importance of the sum of the distances from the origin to the foci, which simplifies the problem without requiring the full equation of the ellipse.

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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!
 
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Hi hellboydvd ! Welcome to PF! :smile:

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

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