Question

Find an equation of the line containing the centers of the two circles whose equations are given below.x2+y2- 4x+ 10y+4 = 0 x2+y2+10x+4y+25 = 0 The equation of the line is (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Answer 1

The equation of the line is
`y = -x-3`.

Explanation:

Let be
`x^(2)+y^(2)-4\cdot x +10\cdot y +4 = 0` and
`x^(2)+y^(2)+10\cdot x +4\cdot y + 25 = 0` the general equations of the two circles, which must be transformed into their standard forms, which are:

`(x-h)^(2)+(y-k)^(2) = r^(2)`

Where:

`h`,
`k` - Components of the center of the circle, dimensionless.

`r` - Radius of the circle, dimensionless.

1)
`x^(2)+y^(2)-4\cdot x +10\cdot y +4 = 0`
`\wedge`
`x^(2)+y^(2)+10\cdot x +4\cdot y + 25 = 0` Given

2)
`(x^(2)-4\cdot x + 4) + (y^(2)+10\cdot y +25) = 25\,\wedge \,(x^(2)+10\cdot x +25)+(y^(2)+4\cdot y + 4) = 4` Associative property/Compatibility with addition

3)
`(x-2)^(2)+(y+5)^(2) = 25\,\wedge \,(x+5)^(2)+(y-2)^(2) = 4` Perfect square trinomial/Result

The centers of both circles are
`(2,-5)` and
`(-5, 2)`, respectively.

A line is represented by the following expression:

`y-y_(o) = m\cdot (x-x_(o))`

Where:

`m = (y_(2)-y_(1))/(x_(2)-x_(1))`

If
`(x_(1), y_(1)) = (2,-5)`,
`(x_(2),y_(2)) = (-5,2)` and
`(x_(o),y_(o)) = (2,-5)`, the equation of the line is:

`m = (2-(-5))/(-5-2)`

`m = -1`

`y - (-5) = -1 \cdot (x-2)`

`y + 5 = -x +2`

`y = -x-3`

The equation of the line is
`y = -x-3`.