Question

consider the following equation of an ellipse.

25x² + 49y² - 200x + 294y - 384 = 0

step 2 of 4: find the center of this ellipse.

Explanation:

Step1: Group \(x\) and \(y\) terms

Group the \(x\) - terms and \(y\) - terms together:
$$25x^{2}-200x + 49y^{2}+294y=384$$

Step2: Factor out the coefficients of \(x^{2}\) and \(y^{2}\)

Factor out \(25\) from the \(x\) - terms and \(49\) from the \(y\) - terms:
$$25(x^{2}-8x)+49(y^{2}+6y)=384$$

Step3: Complete the square for \(x\) and \(y\)

For the \(x\) - part: \((x - 4)^{2}=x^{2}-8x + 16\). For the \(y\) - part: \((y + 3)^{2}=y^{2}+6y+9\)
$$25(x^{2}-8x + 16)-25\times16+49(y^{2}+6y + 9)-49\times9=384$$

Step4: Simplify the equation

$$25(x - 4)^{2}+49(y + 3)^{2}=384+400 + 441$$
$$25(x - 4)^{2}+49(y + 3)^{2}=1225$$
$$\frac{(x - 4)^{2}}{49}+\frac{(y + 3)^{2}}{25}=1$$

Answer:

The center of the ellipse is \((4,-3)\)