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?

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Answer 1 · 2017-01-04T15:42:13

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

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

#(x - h)^2/a^2 - (y - k)^2/b^2 = 1" [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:

#2a = 2#

#a = 1#

The foci for this type are located at #(h - sqrt(a^2 + b^2), k) and (h + sqrt(a^2 + b^2), k)#

Using the foci, #(2, 2) and (6,2)# and #a = 1#, we can write the following equations:

#h - sqrt(1 + b^2) = 2" [2]"#
#h + sqrt(1 + b^2) = 6" [3]"#
#k = 2" [4]"#

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

#2h = 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]:

#(x - 4)^2/1^2 - (y - 2)^2/sqrt(3)^2 = 1" [2]"#

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