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Write the equation in standard form for the ellipse with center at the origin, vertex (0,6)(0,6), and co-vertex (−5,0)(-5,0).

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Q. Write the equation in standard form for the ellipse with center at the origin, vertex (0,6)(0,6), and co-vertex (−5,0)(-5,0).
  1. Vertex determination: Vertex: (0,6)(0, 6) means the ellipse is vertical cuz the vertex is on the yy-axis.
  2. Center calculation: Center (h,k)(h, k) is (0,0)(0, 0). For vertex (0,6)(0, 6), the value of aa is the distance from the center to the vertex on the y-axis.\newlinea=∣6−0∣=6a = |6 - 0| = 6.
  3. Calculation of aa: Co-vertex: (−5,0)(-5, 0) gives us the value of bb, which is the distance from the center to the co-vertex on the x-axis.b=∣−5−0∣=5.b = \left| -5 - 0 \right| = 5.
  4. Calculation of b: Now we plug in the values for aa and bb into the standard form equation of an ellipse.\newlineThe equation is (x−h)2/b2+(y−k)2/a2=1(x - h)^2/b^2 + (y - k)^2/a^2 = 1.
  5. Equation of ellipse: Substitute h=0h = 0, k=0k = 0, a=6a = 6, and b=5b = 5 into the equation.(x−0)252+(y−0)262=1.\frac{(x - 0)^2}{5^2} + \frac{(y - 0)^2}{6^2} = 1.
  6. Substitution of values: Simplify the equation. x225+y236=1\frac{x^2}{25} + \frac{y^2}{36} = 1.

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