ellipses

[wooden girder]

Can someone please help me solve this? the product of the distances of the two focal points from a tangent is equal to the square of the semi-minor axis

[raven tinsel]

draw a picture

[past marten]

If $P$ is the point on the ellipse and it has foci at $S$ and $S'$, you seek $PS\cdot PS'$. And, if $M$ and $M'$ are the feet of the perpendiculars from $P$ to the directrices, then $PS = ePM$, so the product you seek reduces to $e^2PM\cdot PM'$ where $e$ is the eccentricity. $PM$ and $PM'$ are easily calculated as horizontal distances.

[wet kettle]

Jay

[wooden girder]

what do you mean with if m are the feet of the perpendiculars from P to the directrices?

[past marten]

draw a line through P which is perpendicular to the directrices. That line meets the directrices at points M and M'