Question

Graph the circle (2x - 3)2 + (y + 3)2 =36

Answer 1

`\mathrm{Ellipse\:with\:center}\:\left(h,\:k\right)=\left((3)/(2),\:-3\right),\:\:\mathrm{semi-major\:axis}\:b=6,\:\:\mathrm{semi-minor\:axis}\:a=3`

Explanation:

`\left(2x-3\right)^2+\left(y+3\right)^2=36\\(\left(x-h\right)^2)/(a^2)+(\left(y-k\right)^2)/(b^2)=1\:\mathrm{is\:the\:ellipse\:standard\:equation}\\\mathrm{with\:center}\:\left(h,\:k\right)\:\mathrm{and\:}a,\:b\mathrm{\:are\:the\:semi-major\:and\:semi-minor\:axes}\\\mathrm{Rewrite}\:\left(2x-3\right)^2+\left(y+3\right)^2=36\:\mathrm{in\:the\:form\:of\:the\:standard\:ellipse\:equation}\\\left(2x-3\right)^2+\left(y+3\right)^2=36\\\mathrm{Rewrite\:as}\\\left(2x-3\right)^2+\left(y+3\right)^2-36=0`

`\mathrm{Simplify}\:\left(2x-3\right)^2+\left(y+3\right)^2-36:\quad 4x^2-12x+y^2+6y-18\\4x^2-12x+y^2+6y-18=0\\\mathrm{Add\:}18\mathrm{\:to\:both\:sides}\\4x^2-12x+y^2+6y=18\\Factor\:out\:coefficient\:of\:square\:terms\\4\left(x^2-3x\right)+\left(y^2+6y\right)=18\\\mathrm{Divide\:by\:coefficient\:of\:square\:terms:\:}4\\\left(x^2-3x\right)+(1)/(4)\left(y^2+6y\right)=(9)/(2)\\\mathrm{Divide\:by\:coefficient\:of\:square\:terms:\:}1`

`(1)/(1)\left(x^2-3x\right)+(1)/(4)\left(y^2+6y\right)=(9)/(2)\\\mathrm{Convert}\:x\:\mathrm{to\:square\:form}\\(1)/(1)\left(x^2-3x+(9)/(4)\right)+(1)/(4)\left(y^2+6y\right)=(9)/(2)+(1)/(1)\left((9)/(4)\right)\\\mathrm{Convert\:to\:square\:form}\\(1)/(1)\left(x-(3)/(2)\right)^2+(1)/(4)\left(y^2+6y\right)=(9)/(2)+(1)/(1)\left((9)/(4)\right)\\\mathrm{Convert}\:y\:\mathrm{to\:square\:form}`

`(1)/(1)\left(x-(3)/(2)\right)^2+(1)/(4)\left(y^2+6y+9\right)=(9)/(2)+(1)/(1)\left((9)/(4)\right)+(1)/(4)\left(9\right)\\\mathrm{Convert\:to\:square\:form}\\(1)/(1)\left(x-(3)/(2)\right)^2+(1)/(4)\left(y+3\right)^2=(9)/(2)+(1)/(1)\left((9)/(4)\right)+(1)/(4)\left(9\right)\\\mathrm{Refine\:}(9)/(2)+(1)/(1)\left((9)/(4)\right)+(1)/(4)\left(9\right)\\(1)/(1)\left(x-(3)/(2)\right)^2+(1)/(4)\left(y+3\right)^2=9`

`\mathrm{Divide\:by}\:9\\(\left(x-(3)/(2)\right)^2)/(9)+(\left(y+3\right)^2)/(36)=1\\\mathrm{Rewrite\:in\:standard\:form}\\(\left(x-(3)/(2)\right)^2)/(3^2)+(\left(y-\left(-3\right)\right)^2)/(6^2)=1\\\mathrm{Therefore\:ellipse\:properties\:are:}\\\left(h,\:k\right)=\left((3)/(2),\:-3\right),\:a=3,\:b=6\\b>a\:\mathrm{therefore}\:b\:\mathrm{is\:semi-major\:axis\:and}\:a\:\mathrm{is\:semi-minor\:axis}`

`\mathrm{Ellipse\:with\:center}\:\left(h,\:k\right)=\left((3)/(2),\:-3\right),\:\:\mathrm{semi-major\:axis}\:b=6,\:\:\mathrm{semi-minor\:axis}\:a=3`

Answer 2

Answer: 9 on the x axis and 18 on the y axis