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


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.

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2 Answers 2

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

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  • $\begingroup$ If I find value of 'h' from second equation and substitute into first, will it do? $\endgroup$
    – Bloopy
    Nov 25, 2016 at 14:24
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    $\begingroup$ Yes, it will do. Although you might consider replacing the complete term $(x-h)$ $\endgroup$ Nov 25, 2016 at 14:33
  • $\begingroup$ I would leave the $h$ in there. $\endgroup$
    – tomi
    Nov 25, 2016 at 14:34
  • $\begingroup$ @tomi I guess 'h' being the arbitrary constant will have to be removed. I am not too sure what you meant by your comment. $\endgroup$
    – Bloopy
    Nov 25, 2016 at 15:30
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Rearrange your equation:

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

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

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  • $\begingroup$ This has any half-circle with center $(h,2)$ as solution, not just those with radius $5$. $\endgroup$ Nov 25, 2016 at 17:42

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