Circle Inscribed in a Parabola?

[DakMasterFlash]

Homework Statement

Find the largest circle centered on the positive y-axis which touches the origin and which is above y=x^2

Homework Equations

equation of a circle: r^2=(x-a)^2+(y-b)^2
equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2
equation of a parabola: y=x^2

The Attempt at a Solution

For the life of me, I couldn't figure out how to do this problem. I "eyeballed" the graphs and realized that a circle with a radius of 0.5 perfectly fits the parameters required by the problem statement, but I have no idea how to do this mathematically.

A radius of 0.5 yields the equation:
x^2+(y-0.5)^2=0.5^2
x^2+y^2-y+0.25=0.25
x^2+y^2-y=0

I've tried (rather aimlessly) taking the derivative of this function, yielding:
2x+2yy'-1y'=0

However, I realize that I should use the generic equation for a circle, which gives me:
x^2+(y-b)^2=r^2
x^2+y^2-yb+b^2=r^2

As I'm sure you all can tell, I'm not making much progress as to an actual solution to this problem.
I was hoping that you guys could give me some direction and a hint or two.

Any help would be appreciated! Thanks!

 

[jbunniii]

equation of a circle: r^2=(x-a)^2+(y-b)^2
equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2

So far so good. You are given one additional constraint: the circle must touch the origin. What does that tell you about how $r$ and $b$ are related to each other?

equation of a parabola: y=x^2

OK, can you write down an inequality that describes the region above the parabola?

 

[Dick]

Another way to think about it is that you are looking for the 'osculating circle' to y=x^2 at (0,0). It will have the same curvature as y=x^2 at the origin. y'' must be the same for the two curves there.

 

[utkarshakash]

I have some other kind of approach which uses the concept of projectiles in physics (I know it sounds weird but it works ). Here goes the solution:

Let us assume the parabola to be y= -x^2. I simply did this because trajectory of a projectile is always a concave down parabola and also because changing the equation won't have any effect on the original question. Now let's say the projectile is fired with a velocity of v at an angle Θ from horizontal.
$x= v cos \theta t \\ y = v sin \theta t - gt^2 / 2$

If you eliminate x and y from the equations, you get the eqn of trajectory which is

$y = x tan \theta - \dfrac{g sec^2 \theta}{2 v^2} x^2$

But the original trajectory is y = -x^2. Comparing the two equations

tanΘ = 0 and $\dfrac{g sec^2 \theta}{2 v^2} = 1$

The radius of curvature is given by r= v^2 /g.

Solving the above eqns you can see that r indeed comes out to be 1/2!

But I'd also like to point out that this has no physical significance as Θ can never be zero in a projectile and if it is zero, then it would be wrong to call it a projectile motion. But mathematically I don't see any problem with this approach. Now, it's upto you whether you want to go with this method or not.

Cheers.