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Write the equation in standard form for the ellipse with center at the origin, vertex (−11,0)(-11,0), and co-vertex (0,5)(0,5).

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Q. Write the equation in standard form for the ellipse with center at the origin, vertex (−11,0)(-11,0), and co-vertex (0,5)(0,5).
  1. Center at Origin: Center (h,k)(h, k) is at the origin, so h=0h = 0 and k=0k = 0.
  2. Calculate Semi-Major Axis: The vertex (−11,0)(-11, 0) gives us the length of the semi-major axis, aa, which is the distance from the center to the vertex along the x-axis.\newlinea=∣−11−0∣=11a = |-11 - 0| = 11
  3. Calculate Semi-Minor Axis: The co-vertex (0,5)(0, 5) gives us the length of the semi-minor axis, bb, which is the distance from the center to the co-vertex along the y-axis.\newlineb=∣0−5∣=5b = |0 - 5| = 5
  4. Standard Form of Equation: The standard form of the equation for an ellipse is (x−h)2/a2+(y−k)2/b2=1(x - h)^2/a^2 + (y - k)^2/b^2 = 1.\newlinePlugging in the values for h, k, a, and b, we get:\newline(x−0)2/112+(y−0)2/52=1(x - 0)^2/11^2 + (y - 0)^2/5^2 = 1
  5. Simplify Equation: Simplify the equation: x2121+y225=1\frac{x^2}{121} + \frac{y^2}{25} = 1

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