Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 10 - Analytic Geometry - 10.3 The Ellipse - 10.3 Assess Your Understanding - Page 657: 54


Center: $(1,0)$ Foci: $(1,-2\sqrt 2),(1,2\sqrt 2)$ Vertices: $(1,-3),(1,3)$ See graph

Work Step by Step

We are given the ellipse: $9x^2+y^2-18x=0$ Put the equation in standard form: $9(x^2-2x+1)-9+y^2=0$ $9(x-1)^2+y^2=9$ $\dfrac{9(x-1)^2}{9}+\dfrac{y^2}{9}=1$ $\dfrac{(x-1)^2}{1}+\dfrac{y^2}{9}=1$ The equation is in the form: $\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1$ Identify $h,k,a,b$: $h=1$ $k=0$ $a^2=9\Rightarrow a=\sqrt{9}=3$ $b^2=1\Rightarrow b=1$ Determine $c$: $a^2=b^2+c^2$ $c^2=a^2-b^2$ $c^2=9-1$ $c^2=8$ $c=\sqrt 8=2\sqrt 2$ Determine the center: $(h,k)=(1,0)$ Determine the foci: $(h,k-c)=(1,0-2\sqrt 2)=(1,-2\sqrt 2)$ $(h,k+c)=(1,0+2\sqrt 2)=(1,2\sqrt 2)$ Determine the vertices: $(h,k-a)=(1,0-3)=(1,-3)$ $(h,k+a)=(1,0+3)=(1,3)$ Determine the co-vertices: $(h-b,k)=(1-1,0)=(0,0)$ $(h+b,k)=(1+1,0)=(2,0)$ Graph the ellipse:
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