(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the equation of the locus of midpoints of all chords of length 2 units in the circle with equation [tex]x^2+y^2-2y-3=0[/tex]

2. Relevant equations

[tex]d=\sqrt{x_2-x_1)^2+(y_2-y_1)^2}[/tex]

3. The attempt at a solution

I don't know how to begin solving this problem. All I know is [tex](x_2-x_1)^2+(y_2-y_1)^2=4[/tex], and the variables, [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] satisfy the circle equation.

I transformed the circle equation into the general form ~ [tex]x^2+(y-1)^2=4[/tex] So the circle is centred [tex](0,1)[/tex] and radius 2.

Actually while writing this, I realize the locus of the circle will have the same centre thus, [tex]x^2+(y-1)^2=r^2[/tex], and the perpendicular bisector of a chord in a circle passes through its centre, so I can use pythagoras' theorem:

[tex]c^2=r^2+b^2[/tex]

[tex]4=r^2+(\frac{2}{2})^2[/tex]

[tex]r^2=3, r=\sqrt{3}[/tex]

Therefore, the circle equation is: [tex]x^2+(y-1)^2=3[/tex]

Somehow while trying to explain my problem, I figure it out for myself? Anyway, are there any other methods to solve this problem? I thought it would've involved the distance formula in some way.

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# Homework Help: Circle locus

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