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Write the equation in standard form for the ellipse with center at the origin, vertex (0,9)(0,9), and co-vertex (−8,0)(-8,0).

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Q. Write the equation in standard form for the ellipse with center at the origin, vertex (0,9)(0,9), and co-vertex (−8,0)(-8,0).
  1. Identify Orientation of Ellipse: Identify the orientation of the ellipse.\newlineSince the vertex is (0,9)(0,9) which is on the yy-axis, the ellipse is vertical.
  2. Find Value of 'a': Find the value of 'a'.\newlineThe vertex is (0,9)(0,9), so 'a' is the distance from the center to the vertex along the y-axis.\newlinea=9a = 9
  3. Find Value of 'b': Find the value of 'b'.\newlineThe co-vertex is (−8,0)(-8,0), so 'b' is the distance from the center to the co-vertex along the x-axis.\newlineb=8b = 8
  4. Write Equation in Standard Form: Write the equation of the ellipse in standard form.\newlineThe standard form for a vertical ellipse is (x−h)2/b2+(y−k)2/a2=1(x-h)^2/b^2 + (y-k)^2/a^2 = 1, where (h,k)(h,k) is the center.\newlineSince the center is at the origin (0,0)(0,0), the equation becomes:\newlinex2/b2+y2/a2=1x^2/b^2 + y^2/a^2 = 1\newlineSubstitute ′a′'a' and ′b′'b' into the equation:\newlinex2/82+y2/92=1x^2/8^2 + y^2/9^2 = 1
  5. Simplify the Equation: Simplify the equation. x264+y281=1\frac{x^2}{64} + \frac{y^2}{81} = 1

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