Write the equation of the ellipse in standard form: 4x^2-9y^2-40x+36y+100=0
That is not an ellipse. You cannot have x^2 and y^2 terms of opposite sign, for an ellipse.
You have a hyperbola. Complete the squares for standard form.
okay, Thank you very much:)
wait, I still end up with the wrong answer. I end up with
4(x-5)^2/-120 - 9(y+2)^2=-120
9 can't go into -120
My procedure:
(4x^2-40x)-(9y^2+36y)=-100
4(x^2-10x +4)-9(y^2 +4 +4)=-100+ 4(4)-9(4)
4(x-5)^2-9(y+2)^2=-120
(x-5)^2/-30 - ? 9 can't go into -120 evenly
To write the equation of the ellipse in standard form, we need to rearrange the given equation.
Start by grouping the x-terms and the y-terms separately:
4x^2 - 40x - 9y^2 + 36y + 100 = 0
Now, complete the square for both the x-terms and y-terms. For the x-terms, divide the coefficient of x by 2, square it, and add it to both sides of the equation. For the y-terms, follow the same steps.
4(x^2 - 10x) - 9(y^2 - 4y) + 100 = 0
Let's focus on completing the square for the x-terms:
1. Take the coefficient of x (-10) and divide it by 2: -10/2 = -5.
2. Square the result from step 1: (-5)^2 = 25.
3. Add 25 to both sides of the equation:
4(x^2 - 10x + 25) - 9(y^2 - 4y) + 100 = 4(25) - 9(y^2 - 4y) + 100
4(x - 5)^2 - 9(y^2 - 4y) + 100 = 100 - 36
Now, let's complete the square for the y-terms:
1. Take the coefficient of y (-4) and divide it by 2: -4/2 = -2.
2. Square the result from step 1: (-2)^2 = 4.
3. Add 4 to both sides of the equation:
4(x - 5)^2 - 9(y^2 - 4y + 4) + 100 = 100 - 36 + 36
4(x - 5)^2 - 9(y - 2)^2 + 100 = 100
Simplify the equation:
4(x - 5)^2 - 9(y - 2)^2 = 0
Now, divide every term by 100 to obtain the standard form, where both fractions are 1:
(x - 5)^2/25 - (y - 2)^2/11.11 = 1
So, the equation of the ellipse in standard form is:
(x - 5)^2/25 - (y - 2)^2/11.11 = 1