Question

If x – y = 2, then point (x, y) is equidistant from (7, 1) and ____?
$
  {\text{A}}{\text{. }}\left( {3,5} \right) \\
  {\text{B}}{\text{. }}\left( {5,3} \right) \\
  {\text{C}}{\text{. }}\left( {6,2} \right) \\
  {\text{D}}{\text{. }}\left( {2,6} \right) \\
$

Answer 1

Hint: To find the point equidistant from (a, b) and (7, 1), we first compute the line equation of the line perpendicular to the given line, x – y = 2 using the concept of products of slopes of perpendicular lines is equal to -1. Then we find the point of intersection of these two lines. This is the midpoint of (a, b) and (7, 1). We compute the values of a and b using the midpoint formula.

Complete step-by-step answer:
Given Data,
Line x – y = 2 -- (1)
Point (7, 1) and (a, b)
We know the slope of the line of the form, y = mx + c is ‘m’.
The given line equation is x – y = 2, rearranging this line equation in the above form we get,
Y = x – 2, therefore the slope of this line m = 1.
We know the product slopes of two lines perpendicular to each other is ‘-1’.
Therefore the slope of line line perpendicular to the given line x – y = 2 is given by,
Let the slope of this line be k
⟹1 × k = -1
⟹k = - 1
Hence the equation of this line perpendicular to the given line x – y = 2 having a slope -1 is given by,
⟹y = (-1) x + c
⟹x + y = c
This line passes through the point (7, 1), hence it should satisfy the line equation,
⟹7 + 1 = c
⟹c = 8
Therefore the equation of this line is given by, x + y = 8 -- (2)

Now we have to find the point of intersection of these both lines, therefore
From equation (1), x = y + 2
From equation (2), x = 8 – y
⟹y + 2 = 8 – y
⟹2y = 6
⟹y = 3
Therefore x = 3 + 2 = 5
The point of intersection of both these perpendicular lines is (5, 3)
Now this point of intersection will be a midpoint of the line joining the points (7, 1) and (a, b) as it is equidistant from both these points.
We know the formula of midpoint of two points of the form (a, b) and (c, d) is given by,
Midpoint = \[\left( {\dfrac{{{\text{a + c}}}}{2},\dfrac{{{\text{b + d}}}}{2}} \right)\]
We have the midpoint of the points (7, 1) and (a, b) is (5, 3).
\[
   \Rightarrow 5 = \dfrac{{{\text{a + 7}}}}{2},3 = \dfrac{{{\text{b + 1}}}}{2} \\
   \Rightarrow 10 = {\text{a + 7, 6 = b + 1}} \\
   \Rightarrow {\text{a = 3, b = 5}} \\
\]
Therefore the point (x, y) is equidistant from (7, 1) and (3, 5).
Option A is the correct answer.

Note – In order to solve this type of problems the key is to know the formula of a general equation of a line equation, the product of slopes property of two perpendicular lines and the formula of the midpoint of a line joining two points.
The most important step in solving this problem is to understand very clearly what is being asked in the question, we found the point (x, y) which satisfies the given equation by finding the line equation of its perpendicular line and solving these both lines. Then we applied the midpoint formula for this point of intersection and (7, 1) to find the answer.