Question

Relation between x and y such that the point (x, y) is equidistant from the points (-7, -1) and (3, 5) is :

• A. \(5x + y = 2\)
• B. \(5x + 3y = -4\)
• C. \(x - y = 2\)
• D. \(x - 2y = 2\)

Hint:

The locus of a point equidistant from two points is the perpendicular bisector of the segment joining them.
The midpoint of \(A(-7, -1)\) and \(B(3, 5)\) is:
\[ M = \left(\frac{-7+3}{2}, \frac{-1+5}{2}\right) = (-2, 2) \]
Since the midpoint must lie on this line, substitute \(x = -2, y = 2\) into the options:
For (B): \(5(-2) + 3(2) = -10 + 6 = -4\). This matches, confirming Option (B) is correct.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from Coordinate Geometry, focusing on the concept of equidistance.
We need to find the algebraic relationship between the coordinates \(x\) and \(y\) of a point that is located at equal distances from two given points.

Step 2: Key Formula or Approach:
The distance \(d\) between two points \(P(x, y)\) and \(Q(x_1, y_1)\) is given by the distance formula:
\[ d = \sqrt{(x - x_1)^2 + (y - y_1)^2} \]
If \(P(x, y)\) is equidistant from \(A(-7, -1)\) and \(B(3, 5)\), then:
\[ PA = PB \implies PA^2 = PB^2 \]

Step 3: Detailed Explanation:
Using the squared distance formula to avoid square roots:
For \(PA^2\):
\[ PA^2 = [x - (-7)]^2 + [y - (-1)]^2 = (x + 7)^2 + (y + 1)^2 \]
For \(PB^2\):
\[ PB^2 = (x - 3)^2 + (y - 5)^2 \]
Set them equal:
\[ (x + 7)^2 + (y + 1)^2 = (x - 3)^2 + (y - 5)^2 \]
Expand both sides using the identity \((a \pm b)^2 = a^2 \pm 2ab + b^2\):
\[ (x^2 + 14x + 49) + (y^2 + 2y + 1) = (x^2 - 6x + 9) + (y^2 - 10y + 25) \]
Combine constant terms:
\[ x^2 + y^2 + 14x + 2y + 50 = x^2 + y^2 - 6x - 10y + 34 \]
Subtract \(x^2 + y^2\) from both sides:
\[ 14x + 2y + 50 = -6x - 10y + 34 \]
Bring all variable terms to the left side and constants to the right:
\[ 14x + 6x + 2y + 10y = 34 - 50 \]
\[ 20x + 12y = -16 \]
Divide the entire equation by 4 to simplify:
\[ \frac{20x}{4} + \frac{12y}{4} = \frac{-16}{4} \]
\[ 5x + 3y = -4 \]

Step 4: Final Answer:
The relation between \(x\) and \(y\) is \(5x + 3y = -4\).