Question

Write the equation x2 + 2x +y2 - 6y = 35 in standard form.

Answer 1

Given

`x^2+2x+y^2-6y=35`

Complete the square for x, as shown below

`\begin{gathered} x^2+2x+b=(x+a)(x+a)=x^2+2ax+a^2 \\ \Rightarrow2=2a\Rightarrow a=1 \\ \Rightarrow b=a^2=1 \end{gathered}`

Therefore,

`\begin{gathered} x^2+2x+1+y^2-6y=35+1=36 \\ \Rightarrow(x+1)^2+y^2-6y=36 \end{gathered}`

Completing the square for y,

`\begin{gathered} y^2-6y+b=(x+a)^2=x^2+2ax+a^2 \\ \Rightarrow-6=2a\Rightarrow a=-3 \\ \Rightarrow b=a^2=9 \end{gathered}`

Then,

`(x+1)^2+(y-3)^2=45`

The answer is (x+1)^2+(y-3)^2=45. A circumference in its standard form.