Final answer:
The standard form of the given ellipse equation 8x² + 6y² + 16x – 3y + 4 = 0 is 8(x + 1)² + 6(y - 1/4)² - 10 = 0.
Step-by-step explanation:
To find the standard form of the ellipse equation 8x² + 6y² + 16x – 3y + 4 = 0, we need to rearrange the terms to isolate the variables. First, let's group the x terms and y terms separately:
8x² + 16x + 6y² - 3y + 4 = 0
Next, we'll complete the square for both the x and y terms. For the x terms, divide the coefficient of x by 2 and square it:
(8x² + 16x) + (6y² - 3y) + 4 = 0
8(x² + 2x) + 6(y² - (1/2)y) + 4 = 0
8(x² + 2x + 1) + 6(y² - (1/2)y) + 4 = 8 + 6
8(x + 1)² + 6(y - 1/4)² + 4 - 8 - 6 = 0
Simplifying further:
8(x + 1)² + 6(y - 1/4)² - 10 = 0
Now we have the standard form of the ellipse equation, where the center is (-1, 1/4), the semimajor axis is 1 and the semiminor axis is sqrt(10/6).