Point $P(x, y)$ is equidistant from points $A(5, 1)$ and $B(1, 5)$. Prove that $x = y$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:30 · grounding rag
Model Answer
Given: P(x, y) is equidistant from A(5, 1) and B(1, 5), so PA = PB.
$$PA^2 = (x-5)^2 + (y-1)^2$$
$$PB^2 = (x-1)^2 + (y-5)^2$$
Since PA = PB, we have PA² = PB²:
$$(x-5)^2 + (y-1)^2 = (x-1)^2 + (y-5)^2$$
$$x^2 - 10x + 25 + y^2 - 2y + 1 = x^2 - 2x + 1 + y^2 - 10y + 25$$
$$-10x - 2y = -2x - 10y$$
$$-8x + 8y = 0$$
$$\therefore x = y \quad \textbf{(Proved)}$$
Source: Chapter 7, Section 7.2 – Distance Formula
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Explanation
- The key step is squaring both sides of PA = PB to avoid surds, then expanding and simplifying.
- All $x^2$ and $y^2$ terms cancel, leaving a linear equation that directly gives $x = y$.
- Follow Example 4's method from the textbook exactly — examiners expect this step-by-step algebraic approach.
- Write "Proved" or "Hence proved" at the end to signal completion.