Chapter 12.9, Problem 31E

To determine

To find: the standard form of the given equation and identify the conic section represented by this equation.

Answer to Problem 31E

  $\frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y-5\right)}^2}{25}=1$

Ellipse

Explanation of Solution

Given information:

  $25x^2+4y^2-100x-40y+100=0$

Calculation:

The standard form of the equation can be evaluated as shown below.

  $\begin{array}{c}25x^2+4y^2-100x-40y+100=0\\ 25\left(x^2-4x\right)+4\left(y^2-10y\right)=-100\\ 25\left(x^2-4x+4\right)+4\left(y^2-10y+25\right)=-100+100+100\\ 25{\left(x-2\right)}^2+4{\left(y-5\right)}^2=100\\ \frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y-5\right)}^2}{25}=1\end{array}$

Thus, the standard equation is $\frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y-5\right)}^2}{25}=1$ and it represents an ellipse.