Finding the foci and vertices of an ellipse.How would you find the foci and vertices of the following ellipse: 2 x 2 15 + 8 y 2 45 − 2 3 45 x y = 1 ?

Question

Finding the foci and vertices of an ellipse.How would you find the foci and vertices of the following ellipse:             2              x        2              15    +            8              y        2              45    −            2              3              45    x  y  =  1  ?

Matteo Estes · Accepted Answer

Step 1Notice first of all that your ellipse is centered at   O  =  (  0  ,  0  ) (because if (x,y) belongs to the ellipse, then also (-x, -y) belongs to it).To find the axes of the ellipse, notice that if   P  =  (  x  ,  y  ) is a vertex, then the tangent at P is perpendicular to PO, that is       y    ′    (  y      /    x  )  =  −  1. You can compute y′ by differentiating the equation of the ellipse:            4      15        x  +            16      45        y      y    ′    −                    2                  3                    45        y  −                    2                  3                    45        x      y    ′    =  0  ,whence:       y    ′    =                    2                  3                y        −        12        x                    16        y        −        2                  3                x              .Step 2The above condition       y    ′    (  y      /    x  )  =  −  1 implies then that the coordinates of a vertex are related by:            y      x        =                    −        1        ±        2                    3              .Plugging that into the ellipse equation you can get the coordinates of the vertices, and then of course those of the foci.

seguitzla · Accepted Answer

Step 1Consider a rotation:   x  =  X  cos  ⁡  α  −  Y  sin  ⁡  α  ,    y  =  X  sin  ⁡  α  +  Y  cos  ⁡  αThen your ellipse becomes   6  (  X  cos  ⁡  α  −  Y  sin  ⁡  α      )    2    +  8  (  X  sin  ⁡  α  +  Y  cos  ⁡  α      )    2    −  2      3    (  X  cos  ⁡  α  −  Y  sin  ⁡  α  )  (  X  sin  ⁡  α  +  Y  cos  ⁡  α  )  =  45The term in XY has coefficient   4  cos  ⁡  α  sin  ⁡  α  −  2      3        cos    2    ⁡  α  +  2      3        sin    2    ⁡  α which you want to be vanishing; dividing by       cos    2    ⁡  α we get   2      3        tan    2    ⁡  α  +  4  tan  ⁡  α  −  2      3    =  0 so   tan  ⁡  α  =            −      2      +      4              2              3              =      1          3          or    tan  ⁡  α  =            −      2      −      4              2              3              =  −      3  Step 2So we can take   α  =  π      /    6 and the equation of the ellipse becomes      X    2    (  6      cos    2    ⁡  α  +  8      sin    2    ⁡  α  −  2      3    cos  ⁡  α  sin  ⁡  α  )  +      Y    2    (  6      sin    2    ⁡  α  +  8      cos    2    ⁡  α  +  2      3    cos  ⁡  α  sin  ⁡  α  )  =  45that is   5      X    2    +  9      Y    2    =  45 or             X      2        9    +            Y      2        5    =  1.Find foci and vertices, then use the inverse rotation.

How would you find the foci and vertices of the following ellipse: (2x^2)/15+(8y^2)/45-(2sqrt3)/45xy=1?

Answered question

Answer & Explanation

New Questions in High school geometry