How to find all the critical points to graph x 2 - y 2 + 9 = 0 including vertices, foci and asymptotes?

Question

How to find all the critical points to graph             x      2        -          y      2        +    9    =    0   including vertices, foci and asymptotes?

Natalia Arroyo · Accepted Answer

x      2        -          y      2        +    9    =    0                or                    y      2        -          x      2        =    9   or                    y        2                    3        2              -                  x        2                    3        2              =    1  .The common vertical equation hyperbola is                               (          y          -          k          )                2                    a        2              -                            (          x          -          h          )                2                    b        2              =    1    ;    h    ,    k   being center.Here       h    =    0    ,    k    =    0    ,    a    =    3    ,    b    =    3  . So center is at             (      0      ,      0      )      Vertices are       a    =    3   units from center . Therefore two verticesare at             (      0      ,      3      )                    and                    (      0      ,      -      3      )       , Focii are       c   units from center.            c      2        =          a      2        +          b      2        =    9    +    9    =    18    ∴    c    =    3          2      . Therefore focii are             (      0      ,      3              2            )                    and                    (      0      ,      -      3              2            )      . Hyperbola has two asymptotes that intersect at the center of the hyperbola. The asymptotes pass through the vertices of a rectangle of dimensions       2    a    =    6                and              2    b    =    6   with its center at             (      0      ,      0      )        ∴   slope       ±          a      b        =    ±    1  . Equation of asymptotes are       y    -    0    =    ±    1          (      x      -      0      )                    or              y    =    ±    x   graph{x^2-y^2+9=0 [-10, 10, -5, 5]}

How to find all the critical points to graph x^2 - y^2 + 9 = 0 including vertices, foci and asymptotes?

Answered question

Answer & Explanation

New Questions in Differential Calculus