Find the Points on an Ellipse Furthest Away From (1,0)

[Drakkith]

Homework Statement

Find the points on the ellipse 4x²+y²=4 that are furthest from the point (1,0) on the ellipse.

Homework Equations

Ellipse: y=±√(4-4x²)
Distance Formula: d=√[(x₂-x₁)²+(y₂-y₁)²]

The Attempt at a Solution

The distance from (1,0) for any point on the ellipse should be d=√[(x-1)²+(±√(4-4x²)²].
That simplifies to d=√[(x-1)²+4-4x²]. The ± in front of √(4-4x²) just turns positive because the final value is squared anyways, right?

However, when I graph the distance formula it doesn't match up with my ellipse. According to my solution manual the maximum distance occurs at x=-1/3, but my distance formula graph just keeps increasing. The manual squares d to give s=d² and then graphs s, which does give a maximum value for s at x=-1/3, but I don't understand why they do this and how it works.

Edit: Added negative sign to x=-1/3.

 

[Ray Vickson]

Homework Statement

Find the points on the ellipse 4x²+y²=4 that are furthest from the point (1,0) on the ellipse.

Homework Equations

Ellipse: y=±√(4-4x²)
Distance Formula: d=√[(x₂-x₁)²+(y₂-y₁)²]

The Attempt at a Solution

The distance from (1,0) for any point on the ellipse should be d=√[(x-1)²+(±√(4-4x²)²].
That simplifies to d=√[(x-1)²+4-4x²]. The ± in front of √(4-4x²) just turns positive because the final value is squared anyways, right?

However, when I graph the distance formula it doesn't match up with my ellipse. According to my solution manual the maximum distance occurs at x=1/3, but my distance formula graph just keeps increasing. The manual squares d to give s=d² and then graphs s, which does give a maximum value for s at x=1/3, but I don't understand why they do this and how it works.

You should show your work, because I am in agreement with your solution manual. Presumably, you have made an error somewhere.

 

[Drakkith]

You should show your work,

I thought I did so. What else would you like to see?

 

[vela]

I'm not in agreement with your solution manual. I find the maxima occur at $x=-1/3$. If you plot the ellipse, you should be able to see x=+1/3 doesn't make sense either.

 

[Drakkith]

Whoops, I forgot to put the negative sign in front of the 1/3. You are correct, Vela, it is -1/3.

 

[vela]

The only thing I can think of is that you're not plotting the right function. What you wrote in the OP is fine. Whether you plot $d$ or $d^2$, you should see a maximum at x=-1/3.

 

[certainly]

Another approach that I find much simpler is to take the derivative of the function $d^2$ w.r.t $x$ and let that equal to 0 to find the maxima points.
since the maximum power of $x$ in $d^2$ is 2, taking the derivative will produce a linear equation and you should have your desired solution.
[EDIT:- @Drakkith you being a mentor why does your profile have the label STARTER on it?]

 

[Drakkith]

The only thing I can think of is that you're not plotting the right function. What you wrote in the OP is fine. Whether you plot $d$ or $d^2$, you should see a maximum at x=-1/3.

Dang it, you are correct. I put 4-4X instead of 4-4X² into my function. Now it has a maximum at x=-1/3.
I swear... I checked my work about 5 times and still missed that. Thanks, Vela.

Another approach that I find much simpler is to take the derivative of the function $d^2$ w.r.t $x$ and let that equal to 0 to find the maxima points.
since the maximum power of $x$ in $d^2$ is 2, taking the derivative will produce a linear equation and you should have your desired solution.

Thanks. I'll try that. I suppose it would be a good way to double check my answer.

[EDIT:- @Drakkith you being a mentor why does your profile have the label STARTER on it?]

Because I started this thread.

 

[certainly]

Because I started this thread.

Oh I see... I always thought a starter was someone who was had just created his PF account a week earlier or something like that...

 

[Drakkith]

Oh I see... I always thought a starter was someone who was had just created his PF account a week earlier or something like that...

Nope. It's just an easy way to see who started the thread. I'm really happy Greg added it.