Question

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

Answer 1

Answer:

Standard form: $\left(x-\dfrac{9}{2}\right)^2+(y+5)^2=\dfrac{121}{4}.$

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

Step-by-step explanation:

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

First, form perfect squares:

$(x^2-9x)+(y^2+10y)+15=0,\\ \\\left(x^2-9x+\dfrac{81}{4}\right)-\dfrac{81}{4}+(y^2+10y+25)-25+15=0,\\ \\\left(x-\dfrac{9}{2}\right)^2+(y+5)^2=10+\dfrac{81}{4},\\ \\\left(x-\dfrac{9}{2}\right)^2+(y+5)^2=\dfrac{121}{4}.$

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