Question

The equation of an ellipse is 3x^(2)+2y^(2)+18x+20y+65=0. Rewrite the equation in standard form.

Answer 1

To convert the given general ellipse equation to standard form, we group x and y terms, factor out their coefficients, complete the square for both, and divide by the constant to achieve the equation (x + 3)^2/4 + (y + 5)^2/6 = 1.

Step-by-step explanation:

To rewrite the equation of an ellipse from general form to standard form, we need to complete the square for both the x and y variables. The given equation is 3x^2 + 2y^2 + 18x + 20y + 65 = 0. To make this easier, we have to group the x terms and y terms.

First, let's rearrange the terms:

3x^2 + 18x + 2y^2 + 20y = -65

Next, we factor the coefficients of x^2 and y^2 out of the first and third terms respectively:

3(x^2 + 6x) + 2(y^2 + 10y) = -65

We will now complete the square for each group by adding and subtracting the respective squared half of the coefficient of x and y to each group. Remember to multiply the added terms outside the parentheses by the factor we factored out:

3[(x^2 + 6x + 9) - 9] + 2[(y^2 + 10y + 25) - 25] = -65

3[(x + 3)^2] - 27 + 2[(y + 5)^2] - 50 = -65

3(x + 3)^2 + 2(y + 5)^2 = -65 + 27 + 50

3(x + 3)^2 + 2(y + 5)^2 = 12

Finally, we need to divide everything by 12 to get the standard form of the ellipse equation:

(x + 3)^2/4 + (y + 5)^2/6 = 1

Answer 2

To rewrite the given equation of an ellipse in standard form, we complete the square for both the x and y terms. The equation in standard form is 3(x+3)²/162 + 2(y+5)²/162 = 1.

Step-by-step explanation:

The given equation of an ellipse is 3x²+2y²+18x+20y+65=0. To rewrite this equation in standard form, we need to complete the square for both the x and y terms.

Let's start with the x terms. We can rewrite the equation as follows:

3(x²+6x)+2y²+20y+65=0

Next, we need to add a constant term inside the parentheses that completes the square for x. Half of the coefficient of x, which is 6 in this case, is 3. So we add (3²) = 9 inside the parentheses and subtract 3*9 = 27 from the constant term outside the parentheses. The equation becomes:

3(x²+6x+9)+2y²+20y+65-27=0

Now, let's do the same for the y terms. The coefficient of y, which is 20, has a half of 10. So we add (10²) =100 inside the parentheses and subtract 2*100 = 200 from the constant term outside the parentheses. The equation becomes:

3(x²+6x+9)+2(y²+10y+25)+65-27-200=0

Simplifying the equation further, we get:

3(x+3)²+2(y+5)²-162=0

Finally, we divide the entire equation by the constant on the right side to make it equal to 1. The equation in standard form is:

3(x+3)²/162 + 2(y+5)²/162 = 1