Circle
Internal Tangency
JEE Main
Grade 11
Question:
The number of circles that pass through both $(0, 0)$ and $(1, 0)$, and are internally tangent to $x^2 + y^2 = 9$, is _____.
Step-by-Step Solution
Key Concept: Any circle through $(0, 0)$ and $(1, 0)$ has centre $(1/2, k)$ and radius $r = \sqrt{1/4 + k^2}$. Internal tangency inside $x^2 +y^2 = 9$: $r+d = 3$, i.e., $2r = 3$ giving $r = 3/2$ and $k = \pm \sqrt{2}$ — two circles.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 2