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

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What can you say about the distance between two points on an ellipse whose sum of distances from the foci is constant?In summary, the eccentricity of the conic can be found by manipulating the equation e^2=1-(b^2/a^2), where a and b are related through the distance between the two given points. The sum of the distances of the foci from the origin can also be used to find the eccentricity.
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hellboydvd
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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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Welcome to PF!

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:
 
  • #3
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.
 

1. What is the definition of eccentricity?

Eccentricity is a measure of how "elongated" or "flat" a conic section is. It is the ratio of the distance between the focii to the length of the major axis.

2. How do you find the eccentricity of a conic section?

To find the eccentricity of a conic section, you need to know the coordinates of the focii and the distance between them. Then, you can use the formula e = c/a, where e is the eccentricity, c is the distance between the focii, and a is the length of the major axis.

3. Can you find the eccentricity if the conic section passes through the origin?

Yes, you can find the eccentricity even if the conic section passes through the origin. In this case, the distance between the focii is equal to the length of the major axis, so the eccentricity is equal to 1.

4. What type of conic section has an eccentricity greater than 1?

If the eccentricity of a conic section is greater than 1, it is a hyperbola. A hyperbola is a type of conic section that has two distinct branches that open up in opposite directions.

5. How does the eccentricity affect the shape of a conic section?

The eccentricity determines the shape of a conic section. A smaller eccentricity (closer to 0) indicates a more circular shape, while a larger eccentricity (closer to 1) indicates a more elongated or flattened shape. A eccentricity greater than 1 indicates a hyperbola.

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