Question

Question #de290

Answer 1

#(x-4)^2 +(y+5)^2 =36= 6^2

Explanation:

Rewrite as `x^2 -8x +y^2 +10y +5=0`

`x^2 -8x +16 -16 +y^2 +10y +25-25+5`

`(x-4)^2 +(y+5)^2 -16-25+5=0`

`(x-4)^2 +(y+5)^2 =36= 6^2`

Answer 2

You complete the squares, using the patterns:
`(x - h)^2 = x^2 - 2hx + h^2" [1]"`
and
`(y - k)^2 = y^2 - 2ky + k^2" [2]"`

Explanation:

The standard form for the equation of a circle is:

`(x - h)^2 + (y - k)^2 = r^2" [3]"`

where x and y correspond to any point, `(x,y)`, on the circle, h and k correspond to the center point, `(h,k)`, and r is the radius.

Given: `x^2 + y^2 - 8x + 10y + 5 = 0" [4]"`

Move the x terms together, the y terms together, and the constant to the right:

`x^2 - 8x + y^2 + 10y = -5" [4]"`

We want to make the first 3 terms in equation [4] look like the right side of equation [1] so we insert an `h^2` as the third term but, to keep the equation balanced, we must add `h^2` on the right:

`x^2 - 8x + h^2 + y^2 + 10y = h^2 -5" [5]"`

The first 3 terms of equation [5] look like the right side of equation [1].

We can match the `-2hx` in equation [1] with the `-8x` in equation [5] and write the equation:

`-2hx = -8x`

Find the value of h by dividing both sides of the equation by -2x:

`h = 4`

This means that, in equation [5], we can replace the terms `x^2 - 8x + h^2` with `(x - 4)^2` and replace the `h^2` on the right with 16:

`(x - 4)^2 + y^2 + 10y = 16 -5" [6]"`

We want to make the y terms in equation [6] look like the right side of equation [2} so we add a `k^2` on the left but, to keep the equation balanced, we must add a `k^2` to the right side:

`(x - 4)^2 + y^2 + 10y + k^2 = k^2 + 16 -5" [7]"`

The y terms on the left of equation [7] look like the right side of equation [2].

Match the `-2ky` in equation [2] with the `+10y` in equation [7] and write the equation:

`-2ky = +10y`

Find the value of k by dividing both sides by -2y:

`k = -5`

This means that, in equation [7], we can replace `y^2 + 10y + k^2` with `(y - -5)^2` and the k^2 on the right with 25:

`(x - 4)^2 + (y - -5)^2 = 25 + 16 -5" [8]"`

Simplify the constants on the right:

`(x - 4)^2 + (y - -5)^2 = 36" [9]"`

Write the constant as a square:

`(x - 4)^2 + (y - -5)^2 = 6^2" [10]"`

This is a circle with a radius of 6 and a center at `(4, -5)`