Question

Find the coordinates of a point on the line x + y = 5 which is equidistant from (6, 4) and (5, 2).

Hint:

An alternative conceptual approach is to find the perpendicular bisector of the segment joining \( A(6, 4) \) and \( B(5, 2) \).
Any point equidistant from \( A \) and \( B \) must lie on their perpendicular bisector.
Intersecting the equation of this perpendicular bisector with the given line \( x + y = 5 \) directly gives the same coordinates.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Coordinate Geometry.
We need to find the coordinates of a point \( Q \) lying on the line \( x + y = 5 \).
The point \( Q \) must be equidistant from two given points, \( A(6, 4) \) and \( B(5, 2) \).

Step 2: Key Formula or Approach:
- Since the point \( Q(x, y) \) lies on the line \( x + y = 5 \), we can express its y-coordinate in terms of \( x \):
\[ y = 5 - x \implies Q(x, 5 - x) \]
- The condition "equidistant" means:
\[ QA = QB \implies QA^2 = QB^2 \]
- Use the distance formula to set up and solve a linear equation for \( x \).

Step 3: Detailed Explanation:
1. Let the required point be \( Q(x, y) \). Since \( x + y = 5 \), we have \( y = 5 - x \).
Thus, the coordinates of \( Q \) are \( (x, 5 - x) \).
2. Let the given points be \( A(6, 4) \) and \( B(5, 2) \).
Using the distance formula, write the expressions for \( QA^2 \) and \( QB^2 \):
\[ QA^2 = (x - 6)^2 + ((5 - x) - 4)^2 = (x - 6)^2 + (1 - x)^2 \]
\[ QB^2 = (x - 5)^2 + ((5 - x) - 2)^2 = (x - 5)^2 + (3 - x)^2 \]
3. Equate \( QA^2 \) and \( QB^2 \) since \( QA = QB \):
\[ (x - 6)^2 + (1 - x)^2 = (x - 5)^2 + (3 - x)^2 \]
4. Expand the squared terms on both sides:
\[ (x^2 - 12x + 36) + (1 - 2x + x^2) = (x^2 - 10x + 25) + (9 - 6x + x^2) \]
Simplify both sides:
\[ 2x^2 - 14x + 37 = 2x^2 - 16x + 34 \]
5. Cancel the quadratic term \( 2x^2 \) from both sides:
\[ -14x + 37 = -16x + 34 \]
6. Group the \( x \)-terms on one side and constant terms on the other:
\[ -14x + 16x = 34 - 37 \]
\[ 2x = -3 \]
\[ x = -\frac{3}{2} = -1.5 \]
7. Substitute \( x = -1.5 \) back into our expression for \( y \):
\[ y = 5 - x = 5 - (-1.5) = 6.5 = \frac{13}{2} \]
8. Thus, the coordinates of the point are \(\left(-\frac{3}{2}, \frac{13}{2}\right)\).

Step 4: Final Answer:
The coordinates of the equidistant point on the line are \(\left(-\frac{3}{2}, \frac{13}{2}\right)\) or \((-1.5, 6.5)\).