I have no idea

L

Loki123

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Determine the geometric location of the centers of the circles tangent to the circle x ^ 2 + y ^ 2 = 2ax and the axis Oy.

This is how I see the problem?
IMG_20220227_200327.jpg
However I am the most confused when it comes to figuring out the equation of the blue curve... How do I solve this?
 
Loki123 said:
Determine the geometric location of the centers of the circles tangent to the circle x ^ 2 + y ^ 2 = 2ax and the axis Oy.

This is how I see the problem?
View attachment 31372
However I am the most confused when it comes to figuring out the equation of the blue curve... How do I solve this?
Click to expand...
What have you tried?

Do you see that a point is on the blue curve if its x-coordinate equals a less than its distance from (a, 0)? (Draw in radii to the points of tangency with the axis and the center of the given circle.)
 
Dr.Peterson said:
What have you tried?

Do you see that a point is on the blue curve if its x-coordinate equals a less than its distance from (a, 0)? (Draw in radii to the points of tangency with the axis and the center of the given circle.)
Click to expand...
I am not sure I understand. IMG_20220227_223510.jpg
 
Loki123 said:
I am not sure how to simplify AD = CD + a, it seems already so simple :)
Click to expand...
Hi Loki. That's not the equation to simplify.

Dr.Peterson said:
Write an equation that says AD = CD + a
Click to expand...
What Dr. Peterson suggests above is not "write AD=CD+a". That's the form of the equation he's thinking of.

You're looking for an answer containing symbol r, yes? How might you write an equation in the form above that contains symbol r?

:)

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Dr.Peterson said:
What have you tried?

Do you see that a point is on the blue curve if its x-coordinate equals a less than its distance from (a, 0)? (Draw in radii to the points of tangency with the axis and the center of the given circle.)
Click to expand...
Dr.Peterson said:
Write an equation that says AD = CD + a, and simplify to get the equation of the locus.
Click to expand...
We're looking for an equation of the curve, in terms of x and y (and the constant a) for a point D(x, y). So what are lengths AD and CD in terms of x, y, and a?

I expected you to have seen a locus problem before so you'd have some idea of the goal. Is this your first?
 
Dr.Peterson said:
We're looking for an equation of the curve, in terms of x and y (and the constant a) for a point D(x, y). So what are lengths AD and CD in terms of x, y, and a?

I expected you to have seen a locus problem before so you'd have some idea of the goal. Is this your first.
Click to expand...
I got it, thank you very much. It was easy to solve after I calculated the lenghts.
And yes it is my first, usually i follow a notebook from class so I know the basics of solving it, but for some reason my class just skipped over this part entirely.
 
Since you've solved it, I can now show the extended version of my picture, from which one can see directly that the curve is a parabola (if you've learned the focus and directrix definition):

1646056708735.png​
 
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