Circle p is tangent to the x-axis and the y-axis. if the coordinates of the center are (r, r), find the length of the chord whose endpoints are the points of tangency.

 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.

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