Question:
The number of common tangents that can be drawn to the circles $x^2 + y^2 - 6x = 0$ and $x^2 + y^2 + 6x + 2y + 1 = 0$ is ________
The number of common tangents that can be drawn to the circles $x^2 + y^2 - 6x = 0$ and $x^2 + y^2 + 6x + 2y + 1 = 0$ is ________
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Intersecting circles â 3 common tangents.
Updated On: Apr 26, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Convert to standard form.
Circle 1:
\[
x^2 + y^2 - 6x = 0 \Rightarrow (x-3)^2 + y^2 = 9
\]
Centre \(C_1 = (3,0)\), radius \(r_1 = 3\)
Circle 2:
\[
x^2 + y^2 + 6x + 2y +1 = 0
\Rightarrow (x+3)^2 + (y+1)^2 = 9
\]
Centre \(C_2 = (-3,-1)\), radius \(r_2 = 3\)
Step 2: Distance between centres. \[ d = \sqrt{(6)^2 + (1)^2} = \sqrt{37} \]
Step 3: Compare. \[ |r_1 - r_2| = 0<d<r_1 + r_2 = 6 \] So circles intersect.
Step 4: Conclusion. Number of common tangents = 3
Step 2: Distance between centres. \[ d = \sqrt{(6)^2 + (1)^2} = \sqrt{37} \]
Step 3: Compare. \[ |r_1 - r_2| = 0<d<r_1 + r_2 = 6 \] So circles intersect.
Step 4: Conclusion. Number of common tangents = 3
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