Find a relation between $x$ and y such that P($x$, y) is equidistant from the points A(3, 5) and B(7, 1). Hence, write the coordinates of the points on $x$-axis and y-axis which are equidistant from points A and B.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
Relation between x and y:
Let P(x, y) be equidistant from A(3, 5) and B(7, 1), so PA = PB ⟹ PA² = PB²
$$( x-3)^2+(y-5)^2=(x-7)^2+(y-1)^2$$
$$x^2-6x+9+y^2-10y+25=x^2-14x+49+y^2-2y+1$$
$$8x - 8y = 16$$
$$\boxed{x - y = 2}$$
Point on x-axis: Let the point be (x, 0). Substituting y = 0 in x − y = 2:
$$x = 2 \implies \text{Point is } (2,\ 0)$$
Point on y-axis: Let the point be (0, y). Substituting x = 0 in x − y = 2:
$$y = -2 \implies \text{Point is } (0,\ -2)$$
Source: Chapter 7, Section 7.2 (Distance Formula)
---
Explanation
- The examiner expects PA² = PB² (squaring avoids the square root and simplifies algebra).
- Expand both sides, cancel x² and y², then collect like terms to get x − y = 2 (1 mark).
- For the x-axis point, use form (x, 0) and substitute y = 0 into the relation (1 mark).
- For the y-axis point, use form (0, y) and substitute x = 0 (1 mark).
- Writing the final coordinates clearly earns full marks — don't skip the verification step, but in a 3-mark answer it's optional.