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Write the equation in standard form for the ellipse with vertices (0,12)(0,12) and (0,−12)(0,-12), and co-vertices (−4,0)(-4,0) and (4,0)(4,0).

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Q. Write the equation in standard form for the ellipse with vertices (0,12)(0,12) and (0,−12)(0,-12), and co-vertices (−4,0)(-4,0) and (4,0)(4,0).
  1. Identify Vertical Ellipse: Vertices are (0,12)(0,12) and (0,−12)(0,-12), so this is a vertical ellipse. The center is at (0,0)(0,0) because it's the midpoint of the vertices.
  2. Calculate aa Value: The distance from the center to a vertex is the value of aa. So, a=12a = 12 because the vertex is 1212 units away from the center on the y-axis.
  3. Calculate bb Value: Co-vertices are (−4,0)(-4,0) and (4,0)(4,0), so bb is the distance from the center to a co-vertex. That means b=4b = 4, since the co-vertex is 44 units away from the center on the x-axis.
  4. Plug in Values: Now we plug in the values for 'aa' and 'bb' into the standard form equation of an ellipse. Since it's vertical, the 'aa' value goes under the yy-term. The equation 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.
  5. Substitute into Equation: Substitute h=0h=0, k=0k=0, a=12a=12, and b=4b=4 into the equation. We get (x−0)2/42+(y−0)2/122=1(x-0)^2/4^2 + (y-0)^2/12^2 = 1.
  6. Simplify Equation: Simplify the equation to get x2/16+y2/144=1x^2/16 + y^2/144 = 1. This is the standard form of the ellipse.

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