Question

The diameter of the circle, whose centre lies on the line \(x + y = 2\) in the first quadrant and which touches both the lines \(x = 3\) and \(y = 2\), is _______ .

Hint:

The centre (h, k) is at distance \(|3 - h|\) from x = 3 and \(|2 - k|\) from y = 2; both equal the radius.

Answer 1

Correct Answer: 3

Let the centre be (h, k) with \(h + k = 2\), \(h, k \gt 0\), so \(k = 2 - h\).
The radius is \(r = |3 - h| = |2 - k| = |2 - (2 - h)| = h\).
So \(|3 - h| = h\). Since \(0 \lt h \lt 2\), \(3 - h = h\), so \(h = \frac{3}{2}\) and \(k = \frac{1}{2}\) (first quadrant).
Radius \(= \frac{3}{2}\), diameter \(= 3\).
Answer: 3