Let be the circle with centre at and radius . If is the circle centred at passing through origin and touching the circle externally, then the radius of is equal to
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
Let the radius of T be r. Since T is centred at (0, y) and passes through the origin, r = |y|, so y = r (taking y > 0).
For T to touch C externally, the distance between centres equals the sum of radii:
distance between (0, r) and (1, 1) = r + 1
√(1 + (r−1)²) = r + 1
Squaring: 1 + r² − 2r + 1 = r² + 2r + 1
1 − 2r = 2r (after cancelling r² and 1 from both sides... let me redo)
1 + (r−1)² = (r+1)²
1 + r² − 2r + 1 = r² + 2r + 1
2 − 2r = 2r + 1
1 = 4r
r = 1/4
For T to touch C externally, the distance between centres equals the sum of radii:
distance between (0, r) and (1, 1) = r + 1
√(1 + (r−1)²) = r + 1
Squaring: 1 + r² − 2r + 1 = r² + 2r + 1
1 − 2r = 2r (after cancelling r² and 1 from both sides... let me redo)
1 + (r−1)² = (r+1)²
1 + r² − 2r + 1 = r² + 2r + 1
2 − 2r = 2r + 1
1 = 4r
r = 1/4
[JEE Main 2014]