Question

The distance between the internal and external centres of similitude with respect to the circles \[ x^2+y^2+4x+6y+12=0 \] and \[ x^2+y^2-6x-4y+9=0 \] is:

• A. \(5\sqrt{2}\)
• B. \(\dfrac{20\sqrt{2}}{3}\)
• C. \(16\)
• D. \(5\)

Hint:

For two circles with radii \(r_1\) and \(r_2\) and centre distance \(d\), remember the shortcut \[ \text{Distance between internal and external centres of similitude} = \frac{2r_1r_2d}{|r_1^2-r_2^2|}. \] This avoids finding the centres of similitude individually.

Answer 1

The Correct Option is B

Concept: For two circles having centres \(C_1\) and \(C_2\) and radii \(r_1\) and \(r_2\), the internal and external centres of similitude divide the line joining the centres internally and externally in the ratio of the radii. A very useful result states that if the distance between the centres is \(d\), then the distance between the internal and external centres of similitude is \[ \frac{2r_1r_2d}{r_1^2-r_2^2}. \] Therefore, our task is to determine the centres, radii and the distance between the centres of the given circles and then apply the above formula.

Step 1: Convert the first circle into standard form. The first circle is \[ x^2+y^2+4x+6y+12=0. \] Completing squares, \[ (x^2+4x)+(y^2+6y)+12=0 \] \[ (x+2)^2-4+(y+3)^2-9+12=0 \] \[ (x+2)^2+(y+3)^2=1. \] Hence, \[ C_1=(-2,-3), \qquad r_1=1. \]

Step 2: Convert the second circle into standard form. The second circle is \[ x^2+y^2-6x-4y+9=0. \] Completing squares, \[ (x^2-6x)+(y^2-4y)+9=0 \] \[ (x-3)^2-9+(y-2)^2-4+9=0 \] \[ (x-3)^2+(y-2)^2=4. \] Therefore, \[ C_2=(3,2), \qquad r_2=2. \]

Step 3: Find the distance between the centres. Using the distance formula, \[ d=\sqrt{(3+2)^2+(2+3)^2} \] \[ =\sqrt{5^2+5^2} \] \[ =\sqrt{50} \] \[ =5\sqrt2. \] Thus, \[ C_1C_2=5\sqrt2. \]

Step 4: Apply the formula for the distance between the centres of similitude. Using \[ \text{Distance} = \frac{2r_1r_2d}{r_2^2-r_1^2}, \] we obtain \[ = \frac{2(1)(2)(5\sqrt2)}{4-1} \] \[ = \frac{20\sqrt2}{3}. \]

Step 5: Write the final answer. Therefore, the distance between the internal and external centres of similitude is \[ \boxed{\frac{20\sqrt2}{3}}. \]