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Write the equation in standard form for the ellipse x2+9y2−18y−9=0x^2 + 9y^2 - 18y - 9 = 0.

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Full solution

Q. Write the equation in standard form for the ellipse x2+9y2−18y−9=0x^2 + 9y^2 - 18y - 9 = 0.
  1. Group and Move Terms: Group the yy terms together and move the x2x^2 and constant to the other side.x2+9(y2−2y)=9x^2 + 9(y^2 - 2y) = 9
  2. Complete the Square for yy: Add (22)2=1\left(\frac{2}{2}\right)^2 = 1 inside the parentheses to complete the square for yy.\newlinex2+9(y2−2y+1)=9+9(1)x^2 + 9(y^2 - 2y + 1) = 9 + 9(1)
  3. Simplify the Equation: Simplify the equation. x2+9(y−1)2=18x^2 + 9(y - 1)^2 = 18
  4. Get Standard Form: Divide everything by 1818 to get the standard form of the ellipse. x218+9(y−1)218=1\frac{x^2}{18} + \frac{9(y - 1)^2}{18} = 1
  5. Simplify Fractions: Simplify the fractions. x218+(y−1)22=1\frac{x^2}{18} + \frac{(y - 1)^2}{2} = 1

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