Question

find the center of this ellipse.
25x² + y² - 100x - 2y + 76 = 0
center : (?, )

Explanation:

Step1: Group the x - terms and y - terms

$$25x^{2}-100x + y^{2}-2y=-76$$

Step2: Factor out the coefficient of \(x^{2}\) from the x - terms

$$25(x^{2}-4x)+(y^{2}-2y)=-76$$

Step3: Complete the square for the x - terms and y - terms

For \(x^{2}-4x\), using \((a - b)^2=a^{2}-2ab + b^{2}\), here \(a = x\), \(2ab=4x\Rightarrow b = 2\), so \(x^{2}-4x=(x - 2)^{2}-4\).
For \(y^{2}-2y\), using \((a - b)^2=a^{2}-2ab + b^{2}\), here \(a = y\), \(2ab = 2y\Rightarrow b = 1\), so \(y^{2}-2y=(y - 1)^{2}-1\).
Substitute back:
$$25((x - 2)^{2}-4)+((y - 1)^{2}-1)=-76$$
$$25(x - 2)^{2}-100+(y - 1)^{2}-1=-76$$

Step4: Simplify the equation

$$25(x - 2)^{2}+(y - 1)^{2}=25$$
Divide by 25:
$$\frac{(x - 2)^{2}}{1}+\frac{(y - 1)^{2}}{25}=1$$

Answer:

\((2,1)\)