Chapter 9.6, Problem 25PPS

To determine

To Explain: The given equation is parabola, hyperbola, circle or ellipse.

Answer to Problem 25PPS

Circle ${\left(x-\frac{3}{2}\right)}^2+{\left(y-\frac{9}{4}\right)}^2=\frac{117}{16}$ .

Explanation of Solution

Given:

  $4x^2+4y^2-12x-18y=0$

Formula Used:

  $\begin{array}{l}A{x}^2+Bxy+c{y}^2+Dx+Ey+F=0\\ if\\ B^2-4AC<0Ellipse\\ ifA=CCircle\\ B^2-4AC=0Parabola\\ B^2-4AC>0Hyperbola\end{array}$

Calculation:

  $\begin{array}{l}4x^2+4y^2-12x-18y=0\\ A=4\\ B=0\\ C=4\\ B^2-4AC=0-4\left(4\right)\left(4\right)=-64<0\\ A=C\\ 4x^2+4y^2-12x-18y=0\\ \Rightarrow 4\left(x^2-3x+\frac{9}{4}\right)+4\left(y^2-\frac{18}{4}y+\frac{81}{16}\right)=9+\frac{81}{4}=\frac{117}{4}\\ \Rightarrow {\left(x-\frac{3}{2}\right)}^2+{\left(y-\frac{9}{4}\right)}^2=\frac{117}{16}\end{array}$

Conclusion:

Thus , the given equation is of circle ${\left(x-\frac{3}{2}\right)}^2+{\left(y-\frac{9}{4}\right)}^2=\frac{117}{16}$ .