Answer: The length of the chord AB is √2 r units.
Step-by-step explanation: Given in the question that Circle P is tangent to the X-axis at the point A and tangent to the Y-axis at the point B.
Also, the co-ordinates of centre of the circle P are C(r, r). See the attached image below.
We are to find the length of the chord AB, with end-points A and B as the points of tangency.
From the figure, we note that
the co-ordinates of the points A and B are (0, r) and (r, 0) respectively.
As the X-axis and Y-axis are perpendicular to each other, so the lines parallel to them, OB and OA are also perpendicular to each other.
That is, ∠ACB = 90°.
This implies that the triangle ACB is a right-angled triangle.
Now, using Pythagoras theorem in ΔACB, we have
The lengths of the line segments CA and CB are calculated, using distance formula, as follows:
From equation (i), we get
Thus, the length of the chord AB is √2 r units.