Chapter 12, Problem 1CLT

To determine

Identify the conic and sketch its graph.

Answer to Problem 1CLT

The conic is ellipse.

Explanation of Solution

Given information:

Identify the conic and sketch its graph.

  $\frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y+1\right)}^2}{9}=1$

Calculation:

Consider the given equation,

  $\frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y+1\right)}^2}{9}=1$

To identify conic type rearranges the equation in standard form,

  $\frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y+1\right)}^2}{9}=1$

Now take LCM.

  $9x^2+4y^2-36x+8y+4=36$

Standard form is.

  $A{x}^2+C{y}^2+Dx+Ey+F=0$

  $9x^2+4y^2-36x+8y+4=0$

On comparing with standard form the value of ,

  $\begin{array}{l}A=9\\ C=4\end{array}$

To identify the conic type consider these conditions,

  $\begin{array}{cc}\begin{array}{l}conic\\ type\end{array}& condition\\ Circle& A=C\\ Parabola& AC=0\\ Ellipse& AC>0\\ Hyperbola& AC<0\end{array}$

Now

  $\begin{array}{l}A=9\\ C=4\end{array}$

Both values are of same sign and greater than zero, so the given equation represents as conic type is ellipse.

Now draw ellipse of equation, $\frac{{\left(x-2\right)}^2}{4}+\frac{{\left(y+1\right)}^2}{9}=1$

  

Hence the conic is ellipse.