Mathematics

Problem

Point

A

has coordinates

(p,q)

where

p<0

and

q>0

. Another point

B

is defined by interchanging and changing the sign of one of

A

's coordinates, so that

B(q,-p)

. Answer the following questions without assigning specific numerical values to

p

and

q

. a. (i) Determine the quadrant in which point

A

lies and justify your answer. b. (ii) Show logically, using the sign of each coordinate, in which quadrant point

B

must lie. Explain every step of your reasoning. c. (iii) If a third point

C(-p,-q)

is considered, compare the locations of

A

and

C

. State whether they lie in the same quadrant, opposite quadrants or on an axis, and justify your statement. d. (iv) The points

A

and

B

are joined to form a line segment. Prove that this segment must necessarily cross one of the coordinate axes. Identify the axis and support your proof with algebraic sign analysis of the end-point coordinates.

Intuitive explanation

The key insight is that swapping and negating coordinates is a composition of transformations: first reflect across the line $y=x$ (which swaps coordinates), then reflect across the $x$ -axis (which negates the $y$ -coordinate). Since $p<0$ and $q>0$ , point $A$ sits in quadrant II, and tracking how these reflections move it reveals exactly where $B$ lands and how the two points relate geometrically. Understanding this transformation relationship lets you answer questions about distances, angles, and symmetries without plugging in numbers.

Intuitive explanation

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