How to decide whether or not the equation below has a circle as its graph If it does, give the center and the radius. If it does not, describe the graph 25 x 2 + 25 y 2 - 30 x + 30 y - 18 = 0 ?

Question

How to decide whether or not the equation below has a circle as its graph If it does, give the center and the radius. If it does not, describe the graph       25          x      2        +    25          y      2        -    30    x    +    30    y    -    18    =    0  ?

Aubree Phelps · Accepted Answer

Equation of the General Second Degree in             ℝ      2             a          x      2        +    2    h    x    y    +    b          y      2        +    2    g    x    +    2    f    y    +    c    =    0   will represent a Circle , if,            (      i      )        :    a    =    b    ≠    0    ,          (      i      i      )        :    h    =    o    ,    &amp;amp;    ,          (      i      i      i      )        :                  g        2            +                      f        2            -        a    c    &amp;gt;    0    .  In the event, its Centre is             (      -              g        a            ,      -              f        a            )       and Radius is                                                         g              2                        +                                              f              2                        -                    a          c                            |        a        |            .In our Example,             (      i      )        :    a    =    25    =    b    ≠    0    ,          (      i      i      )        :    h    =    0    ,   and,             (      i      i      i      )        :    g    =    -    15    ,    f    =    15    ,    c    =    -    18        ⇒                  g        2            +                      f        2            -        a    c    =    225    +    225    =    900    &amp;gt;    0  .   So, the eqn. represents a circle having centre   (  15      /    25  ,  −  15      /    25  ), i.e.,             (              3        5            ,      -              3        5            )                     &amp;amp; radius                             900                    |        25        |              =          30      25        =          6      5      .

How to decide whether or not the equation below has a circle as its graph...

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