Question

The number of common tangents to the circles \[ x^2+y^2-4x-6y-12=0,\quad x^2+y^2+6x+18y+26=0 \] is:

• A. \( 2 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 5 \)

Hint:

Common tangents: \begin{itemize} \item \( d > r_1+r_2 \Rightarrow 4 \) \item \( d = r_1+r_2 \Rightarrow 3 \) \item \( |r_1-r_2| < d < r_1+r_2 \Rightarrow 2 \) \end{itemize}

Answer 1

The Correct Option is C

Concept: Number of common tangents depends on distance between centers. Step 1: Find centers and radii. Circle 1: \[ (x-2)^2+(y-3)^2=25 \Rightarrow C_1=(2,3), r_1=5 \] Circle 2: \[ (x+3)^2+(y+9)^2=64 \Rightarrow C_2=(-3,-9), r_2=8 \] Step 2: Distance between centers. \[ d=\sqrt{(5)^2+(12)^2}=13 \] Step 3: Compare values. Since: \[ d = r_1 + r_2 = 13 \] Circles are externally tangent. Step 4: Common tangents. Externally touching circles have 3 common tangents.