
minimisation question
Suppose that $\displaystyle a\geq 0$. Find the least distance from the point [tex](a,0) to the ellipse $\displaystyle \frac{x^2}{4}+\frac{y^2}{1}=1$.
Well basically i used the distance formula, so $\displaystyle D=\sqrt{(xa)^2+y^2}=\sqrt{(xa)^2+1\frac{x^2}{4}}$.
Then i set up a function $\displaystyle f(x)=(xa)^2+1\frac{x^2}{4}$.
I differentiated it it find $\displaystyle f'(x)=2(xa)\frac{x}{2}$.
I then found $\displaystyle x=\frac{4a}{3}$ to be the minimum turning point and subbed it back into distance formula to get $\displaystyle D=\sqrt{1\frac{a^2}{3}}$.
However the answer is:
least distance $\displaystyle =\begin{cases}
&\sqrt{1\frac{a^2}{3}} \text{ if } 0\leq a\leq 3/2 \\
&\left  a2 \right  \text{ if } a> 3/2
\end{cases}$
i get why its a2 if a>2 but i dont get why its a2 between a=3/2 and a=2.
Can someone please explain why?

Setting sqrt(1a^2/3) = 2 a
we get a = 3/2 this is the point where the point distance from(a,0) to a point on the ellipse is the same as the distance from (a,0) to the vertex(2,0)
In fact for a = sqrt(3) sqrt(1a^2/3) = 0 and sqrt(1a^2/3) is imaginary
for a > sqrt(3)
If you want I made an animation demonstrating this on my web site:
Scratch Paper

hm...i see where you're getting at, but how come the distance formula stops working between a=3/2 and a=sqrt(3). It's still defined between these values. Also, if you let a=1.6 (say), then sqrt(1 (1.6^2 /3)) < 1.62 . Shouldnt this be the least distance in this case? And also, how come the distance formula gives 0 when a=sqrt(3). Surely, theres no point on the ellipse that gives this distance.

Note x= 4a/3 at the min pt
and y= sqrt(1x^2/4)
if x = 4a/3 y = sqrt[1  (4a^2)/9]
If a = 3/2 x = 2 and y = 0 you are at the vertex
if a > 3/2 you are no longer on the ellipse and the formula
D =sqrt(1a^2/3) is no longer valid as you are not on the ellipse
you're right as sqrt(3) > 3/2 you don't have to worry about a point on the ellipse whose distance is 0
thats also why you don't have to worry about a= 1.6 either

thanks for clarifying that :)

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From a=1.5 to 2, the shortest distance to the ellipse is 2a, or a2 if a>2.