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Write the equation in standard form for the ellipse with vertices (−3,0)(-3,0) and (3,0)(3,0), and co-vertices (0,2)(0,2) and (0,−2)(0,-2).

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Q. Write the equation in standard form for the ellipse with vertices (−3,0)(-3,0) and (3,0)(3,0), and co-vertices (0,2)(0,2) and (0,−2)(0,-2).
  1. Identify Major Axis: Vertices are (−3,0)(-3,0) and (3,0)(3,0), so the major axis is horizontal. The distance from the center to a vertex is the value of aa. Since the vertices are 33 units away from the center, a=3a = 3.
  2. Identify Minor Axis: Co-vertices are (0,2)(0,2) and (0,−2)(0,-2), so the minor axis is vertical. The distance from the center to a co-vertex is the value of bb. Since the co-vertices are 22 units away from the center, b=2b = 2.
  3. Find Center: The center (h,k)(h,k) is at the midpoint of the vertices, which is (0,0)(0,0) because the vertices are equidistant from the origin along the xx-axis.
  4. Standard Form Equation: The standard form of the equation for an ellipse with a horizontal major axis is (x−h)2/a2+(y−k)2/b2=1(x-h)^2/a^2 + (y-k)^2/b^2 = 1. Plugging in the values for hh, kk, aa, and bb, we get (x−0)2/32+(y−0)2/22=1(x-0)^2/3^2 + (y-0)^2/2^2 = 1.
  5. Simplify Equation: Simplify the equation to get x2/9+y2/4=1x^2/9 + y^2/4 = 1. This is the standard form of the equation for the ellipse.

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