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
Please, any help?
for the last one,
i hope you know for an ellipse $\displaystyle c^2+b^2=a^2$.
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:
$\displaystyle (c^2/a^2)+(y^2/b^2)=1$.
using $\displaystyle c^2+b^2=a^2$, we get $\displaystyle y=b^2/a$.
thus distance of point from (c,0) i.e. a focus, is $\displaystyle b^2/a$ which is the semi-latus rectum.
in case u dont know $\displaystyle c^2+b^2=a^2$.
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, $\displaystyle c^2+b^2=a^2$.
Hi
Here is for question vii
http://www.mathhelpforum.com/math-he...tml#post464638
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 $\displaystyle t = \pi/2$. 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 $\displaystyle SG = eMP$ (for that particular point), so that $\displaystyle SG \ne e^2MP$.
For a general point P, the ratio SG/MP will not be constant. The simplest formulas I can get are $\displaystyle SG = ae(1-e\cos t)$ and $\displaystyle MP = \frac{a(1-\cos t)\sqrt{1-e^2\cos^2t}}{\sin t}$, from which you can see that the ratio SG/MP depends on t.