Show that the product of the lengths of the
perpendicular segments drawn from the
foci to any tangent line to the ellips

1 thought on “Show that the product of the lengths of the<br />perpendicular segments drawn from the<br />foci to any tangent line to the ellips”

1. Answer:

   +

   16

   y

   2

   =1e=

   5

   41

   ae=

   41

   Foci S(ae,o)=S(41,o)S(ae,o)=S(

   41

   ,o)

   y=mx+

   a

   2

   m

   2

   b

   2

   y=mx+

   25m

   2

   −16

   P

   1

   =

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   m(

   41

   )+(−1)(o)+

   25m

   2

   −16

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   =

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   25m

   2

   −16

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   P

   0

   =

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   m(−

   41

   )+(−1)(o)+

   25m

   2

   −

   16

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   =

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   25m

   2

   −16

   −

   41

   m

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   P

   1

   P

   2

   =

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   25m

   2

   −16

   +

   41

   m

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   ×

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   25m

   2

   −16−

   41

   m

   ∣

   ∣

   ∣

   ∣

   ∣

   ∣

   =

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   25m

   2

   −16−41m

   2

   ∣

   ∣

   ∣

   ∣

   ∣

   =

   ∣

   ∣

   ∣

   ∣

   ∣

   m

   2

   +1

   −16m

   2

   −16

   ∣

   ∣

   ∣

   ∣

   ∣

   =16

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