- #1

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- Homework Statement
- Find the coordinates at the point on the curve ##y^2={\frac 5 2}(x+1)## which is nearest to the origin.

- Relevant Equations
- The distance from the origin to any point on the graph must be

$$\sqrt{(x-0)^2+(y-0)^2}=\sqrt{x^2+y^2} $$

We can ignore the square root and concentrate on the expression within, and we can express ##y## in terms of ##x##.

$$

x^2+{\frac 5 2}(x+1)

$$

**The Attempt at a Solution**

I know the answer is supposed to be ##(-1,0)##.

However when I differentiate the above expression I get.

$$

2x+{\frac 5 2}

$$

Then the shortest distance would be when the expression equates to 0.

$$

2x+{\frac 5 2}=0

$$

I should be getting ##x=-1## but solving for ##x## I get

$$

x={\frac {-5} 4}

$$

Not only is this not ##-1## , it's not even on the graph of ##y^2={\frac 5 2}(x+1)##.

I've tried checking my steps.

Putting ##x=-1## in ##\sqrt{x^2+{\frac 5 2}(x+1)} ## yields ##1##, which is true so I assume at least up to this step I've done things correctly.

But for ##2x+{\frac 5 2}## , when I put ##x=-1##, I get ##{\frac 1 2}## when I'm supposed to get 0.

So now I'm thinking that my mistake is with how I differentiated ##x^2+{\frac 5 2}(x+1)##.

But I think I followed the correct steps in differentiating it.

What am I missing here?