# SOLUTION: If the ellipse defined by the equation 16x^2+4y^2+96x-8y+84=0 is translated 6 units down and 7 units to the left, write the standard equation of the resulting ellipse

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 Question 625240: If the ellipse defined by the equation 16x^2+4y^2+96x-8y+84=0 is translated 6 units down and 7 units to the left, write the standard equation of the resulting ellipse Answer by Edwin McCravy(8921)   (Show Source): You can put this solution on YOUR website!```16x² + 4y² + 96x - 8y + 84 = 0 Get the constant term off the left side by adding -84 to both sides 16x² + 96x + 4y² - 8y = -84 Swap the two middle terms 16x² + 96x + 4y² - 8y = -84 Factor out the coefficients of x² and y² 16(x² + 6x) + 4(y² - 2y) = -84 To complete the square inside the first parenhtheses, 1. Multiply the coefficient of x, which is +6 by 1/2, getting 3 2. Square this result, (3)² = +9 3. Add +9 inside the first parentheses 4. Multiply +9 by the coefficient we factored out, 16, getting +144 5. Add +144 to the right side 16(x² + 6x + 9) + 4(y² - 2y) = -84 + 144 To complete the square inside the second parenhtheses, 1. Multiply the coefficient of y, which is -2 by 1/2, getting -1 2. Square this result, (-1)² = +1 3. Add +9 inside the first parentheses 4. Multiply +1 by the coefficient we factored out, 4, getting +4 5. Add +4 to the right side 16(x² + 6x + 9) + 4(y² - 2y + 1) = -84 + 144 + 4 Factor the 1st parentheses: x² + 6x + 9 = (x + 3)(x + 3) = (x + 3)² Factor the 2nd parentheses: y² - 2x + 1 = (y - 1)(y - 1) = (y - 1)² Combine terms on the right: -84 + 144 + 4 = 64 16(x + 3)² + 4(y - 1)² = 64 Get a 1 on the right side by dividing through by 64 + = + = 1 + = 1 To translate this equation 6 units down and 7 units to the left, replace y by (y+6) and x by (x+7) + = 1 + = 1 + = 1 Edwin```