Focal Chords of an Ellipse

Geometry Level 3

Suppose A A , B B , C C , and D D are points on an ellipse such that segments A B AB and C D CD intersect at a focus F F .

Given that A F = 3 AF = 3 , C F = 4 CF = 4 , and B F = 5 BF = 5 , what is D F ? DF?

45 23 \frac{45}{23} 30 11 \frac{30}{11} 60 17 \frac{60}{17} 15 4 \frac{15}{4}

Sameer Kailasa by Sameer Kailasa , 25, USA

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1 solution

Atique Shahriar
Oct 11, 2018

Let DF=x;for F point the distance from directrix is f For B point which is 5/e;for A point=3/e;C point =4/e ;

AB/AF=(5/e-3/e)/(f-3/e) =8/3 solve this you will find f=15/4e CD/CF=(x/e-4/e)/(f-4/e) =(x+4)/4 solve this you will find x=60/17 ans!

2 Helpful 2 Interesting 4 Brilliant 4 Confused

Since it takes 5 points, no 3 of which are collinear, to determine a conic, there is no way to definitively answer this question. The diagram implies that the ellipse has a horizontal orientation, but that is not made explicit and cannot be assumed. As it stands, DF can have an arbitrary length, such that A, B, C, and D lie on the ellipse.

Robert Butterfield - 1 year, 11 months ago

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