Question

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

Answer 1

Hint: since there are four different types of conic section, we can relate the given information to that of a hyperbola. We can check it by using all the values given and find all the values of the unknowns in the formula and check what the final answer turns out to be.

Complete step by step solution:
we can relate the given information to that of the hyperbola.
Hence we will find if it matches our assumption.
Now since the foci are oriented horizontally, we will use the standard form for a horizontal transverse axis which is
 $ \dfrac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} - \dfrac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1 - - - \left( 1 \right) $
since the distance of the two given points is 2, we can use this value to find the value of a
\[
  2a = 2\; \\
   \Rightarrow a = 1 \;
 \]
Also we know that foci are located at $ \left( {h - \sqrt {{a^2} + {b^2}} ,k} \right) $ and $ \left( {h + \sqrt {{a^2} + {b^2}} ,k} \right) $ respectively.
Using the points (2, 2), (6, 2) and a = 1 in the foci we get
 $
  h - \sqrt {{a^2} + {b^2}} = 2 \\
   \Rightarrow h - \sqrt {1 + {b^2}} = 2 - - - \left( 2 \right) \;
  $
And
 $
  h + \sqrt {{a^2} + {b^2}} = 6 \\
   \Rightarrow h + \sqrt {1 + {b^2}} = 6 - - - \left( 3 \right) \\
  $
and $ k = 2 $ respectively.
Now adding equation (2) and (3) we can get the value of h
 $
  h - \sqrt {1 + {b^2}} + h + \sqrt {1 + {b^2}} = 8 \\
   \Rightarrow 2h = 8 \\
   \Rightarrow h = 4 \;
  $
If we substitute this value in equation (3) we will get the value of b
 $
  h + \sqrt {1 + {b^2}} = 6 \\
   \Rightarrow 4 + \sqrt {1 + {b^2}} = 6 \\
   \Rightarrow \sqrt {1 + {b^2}} = 6 - 4 \\
   \Rightarrow \sqrt {1 + {b^2}} = 2 \;
  $
Squaring on both sides
 $
  1 + {b^2} = 4 \\
   \Rightarrow {b^2} = 3 \\
   \Rightarrow b = \sqrt 3 \;
  $
Substituting all this values in equation (1) we get
 $ \dfrac{{{{\left( {x - 4} \right)}^2}}}{{{1^2}}} - \dfrac{{{{\left( {y - 2} \right)}^2}}}{{{{\sqrt 3 }^2}}} = 1 $
Which is a hyperbola.
So, the correct answer is “ $ \dfrac{{{{\left( {x - 4} \right)}^2}}}{{{1^2}}} - \dfrac{{{{\left( {y - 2} \right)}^2}}}{{{{\sqrt 3 }^2}}} = 1 $ ”.

Note: Since it was given that the difference of the distance is two we assumed it as a hyperbola. There are four different types of conic sections which have their individual unique property from which we can identify them. Conic sections are very important as they are used for studying 3d geometry.