NCERT Solutions for Class 9th Maths Chapter 1 After Fig. 1.4 — Think and Reflect

Book page 7 Updated on2026-09-08

Q1.

What is the x-coordinate of a point on the y-axis?

Answer

x-coordinate of any point on the y-axis = 0

Why it happens: The x-coordinate of a point is defined as its perpendicular distance from the y-axis, taken as positive to the right and negative to the left. A point that lies on the y-axis is at zero distance from it, so its x-coordinate can only be 0. Every such point therefore has the form (0, y) — for example H (0, 4) and G (0, −4.5) in Fig. 1.2.

Tip: Read the rule the other way round too. If a point’s x-coordinate is 0, it must lie on the y-axis. The condition x = 0 is exactly the equation of the y-axis.

Q2.

Is there a similar generalisation for a point on the x-axis?

Answer

Yes, with the roles of the two axes swapped.

y-coordinate of any point on the x-axis = 0
Every point on the x-axis has the form (x, 0)

Why it happens: The y-coordinate measures perpendicular distance from the x-axis. A point sitting on the x-axis is at zero distance from it, so y = 0. B (4.5, 0) and E (−2.9, 0) in Fig. 1.2 are examples. The condition y = 0 is the equation of the x-axis.

Did you know? Putting the two results together: a point lying on both axes must have x = 0 and y = 0, so it is (0, 0). That is a short proof that the two axes meet at exactly one point — the origin.

Q3.

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

Answer

Yes — exactly when x = y.

Two points coincide ⇔ both coordinates match
(x, y) = (y, x) ⇔ x = y and y = x
Both conditions say the same thing, so ⇔ x = y

For example P (4, 4) and Q (4, 4) are the same point. But P (4, 7) and Q (7, 4) are two different points.

Why it happens: (x, y) is an ordered pair: the first entry is reserved for the horizontal shift and the second for the vertical one. Swapping the entries swaps the two shifts, which moves the point — unless the two shifts happen to be equal, in which case swapping changes nothing. Geometrically, swapping the coordinates reflects the point in the line through O that bisects Quadrants I and III (the line y = x), and the only points a reflection leaves fixed are those on the mirror line itself.

Check it yourself: P (4, 7) and Q (7, 4) are both at distance √(16 + 49) = √65 from O. Equal distance from the origin is not enough to make two points coincide — you need both coordinates to agree.

Q4.

If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?

Answer

Yes, the claim is true. Both halves follow from one fact about ordered pairs: (a, b) = (c, d) exactly when a = c and b = d.

If x = y: then (x, y) = (x, x) and (y, x) = (x, x), so the two are the same point.
If (x, y) = (y, x): matching first entries gives x = y. So equality forces x = y.
Together: (x, y) = (y, x) if and only if x = y.
Contrapositive: if x ≠ y, then (x, y) ≠ (y, x).

Why it happens: The second statement is an “if and only if”, so it needs both directions proved, and the first statement is nothing but the contrapositive of one of them. That is why the two sentences in the claim are not really two separate facts — proving the “if and only if” proves the whole claim.

Tip: Watch the direction of an “if and only if” statement. “x = y gives (x, y) = (y, x)” alone would not settle the question — you must also rule out any other way the two pairs could be equal, and that is what the second direction does.

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