# How do you find the equation of the following conic section and identify it given: All points such that the difference of the distance to the points (2,2) and (6,2) equals 2?

Jan 4, 2017

The reference Hyperbola says that you are asking for the equation of a hyperbola.

#### Explanation:

Because the foci are horizontally oriented, we shall use the standard form for a horizontal transverse axis:

${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1 \text{ [1]}$

We are given that the difference of the distance between the two points is 2. The reference tells us that this can be used to find the value of a:

$2 a = 2$

$a = 1$

The foci for this type are located at $\left(h - \sqrt{{a}^{2} + {b}^{2}} , k\right) \mathmr{and} \left(h + \sqrt{{a}^{2} + {b}^{2}} , k\right)$

Using the foci, $\left(2 , 2\right) \mathmr{and} \left(6 , 2\right)$ and $a = 1$, we can write the following equations:

$h - \sqrt{1 + {b}^{2}} = 2 \text{ [2]}$
$h + \sqrt{1 + {b}^{2}} = 6 \text{ [3]}$
$k = 2 \text{ [4]}$

To find the value of h , add equation [2] to equation [3]:

$2 h = 8$

$h = 4$

Substitute 4 for h into equation [3] and then solve for b:

$4 + \sqrt{1 + {b}^{2}} = 6$

$\sqrt{1 + {b}^{2}} = 2$

$1 + {b}^{2} = 4$

${b}^{2} = 3$

$b = \sqrt{3}$

Substitute these values into equation [1]:

${\left(x - 4\right)}^{2} / {1}^{2} - {\left(y - 2\right)}^{2} / {\sqrt{3}}^{2} = 1 \text{ [2]}$

Equation [2] is the answer; it represents a hyperbola.