Question

Write the equation of the ellipse 25 x 2 4 y 2 50 x 16 y − 59 = 0 in standard form

a) (x-1)²/4 + y²/25 = 1
b) x²/25 + (y-2)²/4 = 1
c) (x+1)²/4 + y²/25 = 1
d) x²/4 + (y-2)²/25 = 1

Answer 1

The standard form of the given ellipse equation is found by completing the square for both x and y terms. The correct standard form of the ellipse's equation is option c) (x + 1)²/4 + y²/25 = 1, when we correct the typo in the y term.

Step-by-step explanation:

To write the equation of the ellipse in standard form, we need to complete the square for both the x and y terms in the given equation: 25x^2 + 4y^2 + 50x + 16y - 59 = 0.

First, we group the x terms and the y terms, moving the constant to the other side of the equation:

1. Group x terms and y terms: 25x^2 + 50x + 4y^2 + 16y = 59
2. Factor out the coefficients of the squared terms: 25(x^2 + 2x) + 4(y^2 + 4y) = 59
3. Add and subtract the necessary constants to complete the square within the parentheses for both x and y: 25[(x + 1)^2 - 1] + 4[(y + 2)^2 - 4] = 59
4. Isolate the completed squares and solve for 1: 25(x + 1)^2 + 4(y + 2)^2 - 25 - 16 = 59 => 25(x + 1)^2 + 4(y + 2)^2 = 100
5. Divide by 100 to get the equation in standard form: (x + 1)^2/4 + (y + 2)^2/25 = 1

So the correct standard form of the ellipse's equation is option c) (x + 1)²/4 + y²/25 = 1, when we correct the typo in the y term.