Question

Find the radius of the circle x^2 - 4x + y^2 - 10y = -4

Answer 1

The equation of the circle given by the problem is:

`x^2-4x+y^2-10y=-4​`

Step to find the radius:

Step 1. Rearrange the terms from the greatest exponent to the lowest exponent:

`x^2+y^2-4x-10y=-4`

Step 2. We need for this equation to be equal to 0, so we add 4 to both sides:

`\begin{gathered} x^2+y^2-4x-10y+4=-4+4 \\ x^2+y^2-4x-10y+4=0 \end{gathered}`

Step 3. Compare the last equation with the general equation:

`x^2+y^2+Dx+Ey+F=0`

Comparing the equations we find the values for D, E, and F:

`\begin{gathered} D=-4 \\ E=-10 \\ F=4 \end{gathered}`

Step 4. With the values of D and E, find the values of a and b defined as follows:

`\begin{gathered} a=-(D)/(2) \\ b=-(E)/(2) \end{gathered}`

Substituting D=-4 and E=-10:

`\begin{gathered} a=-((-4))/(2)=(4)/(2)=2 \\ b=-((-10))/(2)=(10)/(2)=5 \end{gathered}`

a=2 and b=5.

Step 5. The formula to find the radius is:

`r=\sqrt[]{a^2+b^2-F}`

Substituting the values of a, b and F:

`r=\sqrt[]{2^2+5^2-4}`

Solving the operations:

`r=\sqrt[]{4+25-4}`
`r=\sqrt[]{25}`
`r=5`

The radius is equal to 5.

Answer: r=5