Question

Write the equation in standard form. Identify the important features of the graph:

x^2+y^2-9x+10y+15=0

Answer 1

Standard form:
`\left(x-(9)/(2)\right)^2+(y+5)^2=(121)/(4).`

This equation represents the circle with the center at the point
`\left((9)/(2),-5\right)` and the radius
`r=(11)/(2).`

Explanation:

Consider expression
`x^2+y^2-9x+10y+15=0.`

First, form perfect squares:

`(x^2-9x)+(y^2+10y)+15=0,\\ \\\left(x^2-9x+(81)/(4)\right)-(81)/(4)+(y^2+10y+25)-25+15=0,\\ \\\left(x-(9)/(2)\right)^2+(y+5)^2=10+(81)/(4),\\ \\\left(x-(9)/(2)\right)^2+(y+5)^2=(121)/(4).`

This equation represents the circle with the center at the point
`\left((9)/(2),-5\right)` and the radius
`r=(11)/(2).`