The centres of those circles which touch the circle, , externally and also touch the -axis, lie on
- A.a circle
- B.an ellipse which is not a circle
- C.a hyperbola
- D.a parabola
Show answer and solution
Correct option: D
Solution
The given circle has centre (4,4) and radius √(16+16+4) = 6.
Let a circle with centre (h,k) and radius |k| (since it touches the x-axis, radius = k, assuming k>0) touch the given circle externally. Then the distance between centres equals sum of radii:
√((h-4)² + (k-4)²) = 6 + k
Squaring: (h-4)² + (k-4)² = (6+k)²
(h-4)² + k² - 8k + 16 = 36 + 12k + k²
(h-4)² = 20k + 20 = 20(k+1) — wait, let me redo:
(h-4)² + k² - 8k + 16 = k² + 12k + 36
(h-4)² = 20k + 20
(h-4)² = 20(k+1)
Replacing h→x, k→y: (x-4)² = 20(y+1), which is a parabola.
Let a circle with centre (h,k) and radius |k| (since it touches the x-axis, radius = k, assuming k>0) touch the given circle externally. Then the distance between centres equals sum of radii:
√((h-4)² + (k-4)²) = 6 + k
Squaring: (h-4)² + (k-4)² = (6+k)²
(h-4)² + k² - 8k + 16 = 36 + 12k + k²
(h-4)² = 20k + 20 = 20(k+1) — wait, let me redo:
(h-4)² + k² - 8k + 16 = k² + 12k + 36
(h-4)² = 20k + 20
(h-4)² = 20(k+1)
Replacing h→x, k→y: (x-4)² = 20(y+1), which is a parabola.
[JEE Main 2016]