1. What is the x-coordinate of a point on the y-axis?
2. Is there a similar generalisation for a point on the x-axis?
3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.
4. If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?
- 1. What is the x-coordinate of a point on the y-axis?
- 2. Is there a similar generalisation for a point on the x-axis?
- 3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.
- 4. If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?
Step-by-step solution
Idea: The x-coordinate is the distance from the y-axis and the y-coordinate is the distance from the x-axis. Two points are the same only if their x-coordinates are equal and their y-coordinates are equal; the order inside the bracket matters.
1. What is the x-coordinate of a point on the y-axis?
- The x-coordinate is the distance from the y-axis. A point on the y-axis is at distance 0 from it, so its x-coordinate is 0. Every such point is (0, y), e.g. H (0, 4) and G (0, −4.5).1 mark
2. Is there a similar generalisation for a point on the x-axis?
- Yes. A point on the x-axis is at distance 0 from the x-axis, so its y-coordinate is 0. Every such point is (x, 0), e.g. B (4.5, 0) and E (−2.9, 0).1 mark
3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.
- P and Q coincide only if their first coordinates are equal (x = y) and their second coordinates are equal (y = x). Both conditions say the same thing: x = y.½ mark
- So yes, but only when x = y: for example P (3, 3) and Q (3, 3) are the same point. If x ≠ y, e.g. P (2, 5) and Q (5, 2), the points are different: one is 2 right and 5 up, the other 5 right and 2 up.½ mark
4. If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?
- (x, y) = (y, x) means first coordinates equal, x = y, and second coordinates equal, y = x. So the two pairs are equal exactly when x = y.½ mark
- If x ≠ y, the first coordinates differ, so the pairs differ. Hence the claim is true. This is why (x, y) is called an ordered pair.½ mark
Check: Try numbers: (2, 5) is 2 right, 5 up; (5, 2) is 5 right, 2 up, a different place ✓. (4, 4) swapped is still (4, 4) ✓.
Answer to write in the exam
1.
x-coordinate = distance from the y-axis = 0
∴ Every point on the y-axis is (0, y).
2.
Yes. y-coordinate = distance from the x-axis = 0
∴ Every point on the x-axis is (x, 0).
3.
Q (y, x) = P (x, y) ⇔ y = x and x = y ⇔ x = y
∴ They coincide only when x = y, e.g. (3, 3); otherwise they are different points.
4.
(x, y) = (y, x) ⇔ x = y and y = x ⇔ x = y
If x ≠ y, the first coordinates differ ⇒ (x, y) ≠ (y, x)
∴ The claim is true.
Common mistakes that cost marks
- Answering part 1 with “y“. The question asks for the x-coordinate of a point on the y-axis, which is always 0.
- Thinking (2, 5) and (5, 2) are the same point because they use the same numbers. Order matters in a coordinate pair.
- In part 3 answering just “no”. The points do coincide in the special case x = y.
How this can come in the exam
If (a − 2, 7) = (5, b + 1), then a + b =
- 13
- 11
- 9
- 1
Show answer
(A) 13
a − 2 = 5 gives a = 7; b + 1 = 7 gives b = 6; a + b = 13.
Assertion (A): (2, −3) and (−3, 2) are different points.
Reason (R): Two ordered pairs are equal only when their first coordinates are equal and their second coordinates are equal.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
The first coordinates 2 and −3 differ, so by R the points are different.
Try one yourself
For what value of k do the points (k + 1, 4) and (4, k + 1) coincide?
Show answer
They coincide when k + 1 = 4, i.e. k = 3; both points are then (4, 4).
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