Differential Equations whose solution is the family of circles with fixed radius and tangent to a common line

Question

Q. The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is?

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My Attempt:

I can write the equation of a random circle, satisfying the condition mentioned in the question, as:

$$(x – h)^2 + (y – 2)^2 = 25$$

Differentiating, with respect to x,

$$2(x – h) + 2(y – 2) \frac{dy}{dx}= 0$$

I am not sure how to proceed further.

Answer 1

Can you eliminate the arbitrary constant $h$ from your system of equations? That will give you the answer.

Answer 2

Rearrange your equation:

$$2(y-2)\frac {dy}{dx}=-2(x-h)$$

$$\frac {dy}{dx}=-\frac{(x-h)}{(y-2)}$$