- #1

Ascendant0

- 57

- 11

- Homework Statement
- Find the values of h, k, and a that make

the circle (x - h)^2 + (y - k)^2 = a^2 tangent to the parabola

y = x^2 + 1 at the point (1, 2) and that also make the second

derivatives d^2y>dx^2 have the same value on both curves there

- Relevant Equations
- Included in my attempts at a solution

I'm at a loss as to how they got to certain steps in the solutions manual. Here's how far I got with this:

Since the circle is tangent to y = x^2 + 1, the slope at (1, 2) is going to be 2, as is the slope of the 2nd derivative of the circle, so then...

The derivative of the circle would be dy/dx = (x - h)/(y - k), and since dy/dx = 2, I can find that h = 5 - 2k

For the 2nd derivative, this is where the manual loses me. They skip like three steps as to how they simplified it, so I don't see how they get their answer. Doing the 2nd derivative, I get something sloppy with a k^2 and multiple answers, whereas in the book, they get a simple k = 9/2 from an equation they give no explanation how they got to

For the 2nd derivative, I have tried both making the (y - k)^(-1), so I can use the product rule, instead of what I feel is a sloppier quotient rule, but tried that one as well. Either way, it gets sloppy at this point, and the solution loses me:

d^2y/dx^2 = (x - h) (-1) (y - k)^(-2) (dy/dx) + (y - k)^(-1)

Of course I can simplify this, but whether the quotient or product rule for derivatives, I end up with a "k^2" as a part of my derivative, so I end up with two different answers. I haven't tried anything with either answer, as I really want to know how they got to their answer in the book, as I know there is a simpler way to do this, but I don't see how they did

In the book, they don't explain, they just go:

"Also, (x - h)^2 + (y - k)^2 = a^2 ----> y'' = (1 + (y')^2)/(k - y)" and give NO explanation how they got the 2nd derivative from it???

I've been racking my brain on this one, and it's driving me nuts. I need to figure out how to simplify that so I don't get a k^2 and two different answers, but I'm stuck. If someone could point me in the right direction here as to what I'm missing, it would be greatly appreciated. I can't accept not being able to do something like this. My brain goes crazy until I can :P

Since the circle is tangent to y = x^2 + 1, the slope at (1, 2) is going to be 2, as is the slope of the 2nd derivative of the circle, so then...

The derivative of the circle would be dy/dx = (x - h)/(y - k), and since dy/dx = 2, I can find that h = 5 - 2k

For the 2nd derivative, this is where the manual loses me. They skip like three steps as to how they simplified it, so I don't see how they get their answer. Doing the 2nd derivative, I get something sloppy with a k^2 and multiple answers, whereas in the book, they get a simple k = 9/2 from an equation they give no explanation how they got to

For the 2nd derivative, I have tried both making the (y - k)^(-1), so I can use the product rule, instead of what I feel is a sloppier quotient rule, but tried that one as well. Either way, it gets sloppy at this point, and the solution loses me:

d^2y/dx^2 = (x - h) (-1) (y - k)^(-2) (dy/dx) + (y - k)^(-1)

Of course I can simplify this, but whether the quotient or product rule for derivatives, I end up with a "k^2" as a part of my derivative, so I end up with two different answers. I haven't tried anything with either answer, as I really want to know how they got to their answer in the book, as I know there is a simpler way to do this, but I don't see how they did

In the book, they don't explain, they just go:

"Also, (x - h)^2 + (y - k)^2 = a^2 ----> y'' = (1 + (y')^2)/(k - y)" and give NO explanation how they got the 2nd derivative from it???

I've been racking my brain on this one, and it's driving me nuts. I need to figure out how to simplify that so I don't get a k^2 and two different answers, but I'm stuck. If someone could point me in the right direction here as to what I'm missing, it would be greatly appreciated. I can't accept not being able to do something like this. My brain goes crazy until I can :P