1. ## Ellipse Problems

Hi Could someone please give me hints for these last few questions I have to do....

I'm really struggling to finish them off before the weekend's over

2. Originally Posted by xwrathbringerx
Hi Could someone please give me hints for these last few questions I have to do....

I'm really struggling to finish them off before the weekend's over

for the last one,
i hope you know for an ellipse $c^2+b^2=a^2$.
where 2c=focal distance, 2b=minor axis length, 2a=major axis length.
So having origin as centre for the ellipse, the point (c,y) lies on the ellipse for some y which we will find.
therefore, placing (c,y) in standard ellipse eqn:
$(c^2/a^2)+(y^2/b^2)=1$.
using $c^2+b^2=a^2$, we get $y=b^2/a$.
thus distance of point from (c,0) i.e. a focus, is $b^2/a$ which is the semi-latus rectum.

in case u dont know $c^2+b^2=a^2$.
from defn. of ellipse sum of distances of a point from two focii of an ellipse is equal to length of major axis. So, for a standard ellipse having origin as centre, for the point (0,b),
dist of (0,b) from (-c,0)=dist of (0,b) from (c,0)=a. [from geometry]
the triangle formed by (0,0),(c,0),(0,b) is right angled. So from Pythagoras, $c^2+b^2=a^2$.

3. Hmm

For (iv), I've got M to be (a, b((1-cost)/sint))

BUT when I plug that into the distance formula to get MP, it gets really long and I can find no way to simplifiy it at all!

Any hints?

4. Originally Posted by xwrathbringerx
For (iv), I've got M to be (a, b((1-cost)/sint))

BUT when I plug that into the distance formula to get MP, it gets really long and I can find no way to simplifiy it at all!
I agree with that, and I think that (iv) is just plain wrong. In fact, you can easily see that it's wrong by taking $t = \pi/2$. Then P is the point (0,b), the tangent at P is horizontal and so M = (a,b). So in that case MP = a. The normal at P is then the y-axis, so G is at the origin and SG = ae. Therefore $SG = eMP$ (for that particular point), so that $SG \ne e^2MP$.

For a general point P, the ratio SG/MP will not be constant. The simplest formulas I can get are $SG = ae(1-e\cos t)$ and $MP = \frac{a(1-\cos t)\sqrt{1-e^2\cos^2t}}{\sin t}$, from which you can see that the ratio SG/MP depends on t.