Given a point ( x 0 , y 0 ) and a radius r, how do you find the set of all circles that have that radius that pass through the point?

Question

Given a point   (      x    0    ,      y    0    ) and a radius r, how do you find the set of all circles that have that radius that pass through the point?

lorienoldf7 · Accepted Answer

Step 1We are interested in circles in the   ⟨  x  ,  y  ⟩ plane, therefore they will have Cartesian equation:  (  x  −      x    C        )    2    +  (  y  −      y    C        )    2    =      r    2  where their radius   r  &amp;gt;  0 is fixed, so it remains to understand how to determine the centers.In particular, given that we want the circles in question all to pass through   (      x    P    ,        y    P    ) and at the same time we have radius   r  &amp;gt;  0 , it&#039;s evident that   (      x    C    ,        y    C    ) must belong to the circle with center   (      x    P    ,        y    P    ) and radius   r  &amp;gt;  0 , that is:      {                                        x            C                    =                      x            P                    +          r                    cos          ⁡          (          u          )                                                  y            C                    =                      y            P                    +          r                    sin          ⁡          (          u          )                              with    u  ∈  [  0  ,    2  π  )    .In this way, we have determined the Cartesian equation of the sheaf of circles obtainable according to the chosen value of   u  ∈  [  0  ,    2  π  )

Given a point ( x 0 </msub> , y 0 </msub> ) and a radi

Answered question

Answer & Explanation

New Questions in Elementary geometry