Suppose A,B,C,D are points on an ellipse such that

[Guest]

Suppose A,B,C,D are points on an ellipse such that the segments AB and CD intersects at a focus F. Given that AF=3, CF=4, and BF=5, what is DF?

[shyamjayakannan]

Use this property of focal chords: Let AB be a focal chord of an ellipse such that it passes through the focus F. Then [tex]AF\times BF=b^2[/tex], where [tex]b[/tex] is the semi - minor axis. So, [tex]AF\times BF=CF\times DF=b^2\Rightarrow DF=\frac{3\times5}{4}=\boxed{\frac{15}{4}}[/tex]