Circles
Circle
nta_abhyas_2025
Grade 11
Question:
Let $C_1$ be circle $x^2 + y^2 + 8x + 8y = 0$, $C_2$ be circle $x^2 + y^2 + 8x - 8y = 0$, $C_3$ be circle $x^2 + y^2 - 8x + 8y = 0$, and $C_4$ be circle $x^2 + y^2 - 8x - 8y = 0$. Now, common tangents between $C_1$ and $C_2$ is 2, common tangents between $C_1$ and $C_3$ is 2, common tangents between $C_1$ and $C_4$ is 3, common tangents between $C_2$ and $C_3$ is 3, common tangent between $C_3$ and $C_4$ is 2, and common tangent between $C_1$ and $C_4$ is 2.
Step-by-Step Solution
Key Concept: The number of common tangents between two circles depends on their relative positions (external, intersecting, or one inside the other).
We count the common tangents for each pair of circles: $C_1$ and $C_2$ share 2 common tangents, $C_1$ and $C_3$ share 2 common tangents, $C_1$ and $C_4$ share 3 common tangents, $C_2$ and $C_3$ share 3 common tangents, $C_2$ and $C_4$ share 2 common tangents, and $C_3$ and $C_4$ share 2 common tangents. Adding these: $2 + 2 + 3 + 3 + 2 + 2 = 14$ common tangents in total.
Correct Answer: 14