Question 5 of 25intermediate🔍 AnalyzeShort Answer2 marks

Does point Q (y, x)Q\,(y,\, x) ever coincide with point P (x, y)P\,(x,\, y)? Justify your answer.

Correct Answer

Yes, point Q (y, x)Q\,(y,\, x) coincides with point P (x, y)P\,(x,\, y) only when x=yx = y.

According to the properties of coordinates in 2-D space, the order of the coordinates is crucial. If x≠yx \neq y, then (x, y)≠(y, x)(x,\, y) \neq (y,\, x), meaning the points PP and QQ will have different positions on the Cartesian plane. For example, point S(3, −5)S(3,\, -5) is in Quadrant IV, while point Q(−5, 3)Q(-5,\, 3) is in Quadrant II, so they do not coincide.

However, if x=yx = y, then the coordinates of both points become identical, i.e., (x, y)=(y, x)(x,\, y) = (y,\, x). In this specific case, PP and QQ represent the exact same point on the plane, and thus they coincide.

∴ Q (y, x)Q\,(y,\, x) coincides with P (x, y)P\,(x,\, y) only when x=yx = y.

Exercise: Think and Reflect | Q: 3 | (Page 7)
For More Understanding

Explanation

The textbook explicitly states that (x, y)=(y, x)(x,\, y) = (y,\, x) if and only if x=yx = y. It also provides the example of S(3, −5)S(3,\, -5) in Quadrant IV and Q(−5, 3)Q(-5,\, 3) in Quadrant II to show that swapping coordinates changes the point's location when x≠yx \neq y. Therefore, Q (y, x)Q\,(y,\, x) coincides with P (x, y)P\,(x,\, y) only under the condition that x=yx = y.

Solution Steps

  1. Step 1: State the condition for coincidence from the text: (x, y)=(y, x)(x,\, y) = (y,\, x) if and only if x=yx = y.

  2. Step 2: Explain that if x≠yx \neq y, then (x, y)≠(y, x)(x,\, y) \neq (y,\, x), giving different points (e.g., S(3, −5)S(3,\, -5) vs Q(−5, 3)Q(-5,\, 3)).

  3. Step 3: Conclude that Q (y, x)Q\,(y,\, x) coincides with P (x, y)P\,(x,\, y) only when x=yx = y.

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