Question

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.

Answer 1

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

$AB^2=CA^2+CB^2~~~~~~~~~~~~~~~~~~~~~(i).$

The lengths of the line segments CA and CB are calculated, using distance formula, as follows:

$CA=\sqrt{(0-r)^2+(r-r)^2}=\sqrt{r^2}=r,\\\\CB=\sqrt{(r-r)^2+(0-r)^2}=\sqrt{r^2}=r.$

From equation (i), we get

$AB^2=r^2+r^2\\\\\Rightrarrow AB^2=2r^2\\\\\Rightarrow AB=\sqrt2r~\textup{units}.$

Thus, the length of the chord AB is √2 r units.

Answer 2

What are the coordinates of the points of tangency? Yes, it's *that* trivial. 
Apply the Pythagorean theorem to get the distance of r √2.