Question

If the circles \( S = x^2 + y^2 - 14x + 6y + 33 = 0 \) and \( S' = x^2 + y^2 - a^2 = 0 \) (where \( a \in \mathbb{N} \)) have 4 common tangents, then the possible number of values of \( a \) is: 

• A. \(13\)
• B. \(5\)
• C. \(14\)
• D. \(2\)

Hint:

For two circles to have 4 common tangents, the distance between the centers must be greater than the sum of the radii.

Answer 1

The Correct Option is D

Step 1: Find the center and radius of the first circle.
The equation of the first circle is S = x² + y² - 14x + 6y + 33 = 0.
Comparing with the general equation x² + y² + 2gx + 2fy + c = 0, we have:
2g = -14 \Rightarrow g = -7
2f = 6 \Rightarrow f = 3
c = 33
Center C₁ = (-g, -f) = (7, -3)
Radius r₁ = \(\sqrt{g^2 + f^2 - c} = \sqrt{(-7)^2 + (3)^2 - 33} = \sqrt{49 + 9 - 33} = \sqrt{25} = 5\)

Step 2: Find the center and radius of the second circle.
The equation of the second circle is S' = x² + y² - a² = 0.
Comparing with the general equation x² + y² + 2gx + 2fy + c = 0, we have:
2g = 0 \Rightarrow g = 0
2f = 0 \Rightarrow f = 0
c = -a²
Center C₂ = (-g, -f) = (0, 0)
Radius r₂ = \(\sqrt{g^2 + f^2 - c} = \sqrt{0^2 + 0^2 - (-a^2)} = \sqrt{a^2} = |a| = a\) (since a \(\in \mathbb{N}\))

Step 3: Find the distance between the centers.
Distance C₁ C₂ = \(\sqrt{(7 - 0)^2 + (-3 - 0)^2} = \sqrt{49 + 9} = \sqrt{58}\)

Step 4: Determine the condition for 4 common tangents.
For two circles to have 4 common tangents, they must be completely outside each other.
This means that the distance between the centers must be greater than the sum of the radii:
C₁ C₂ > r₁ + r₂
\(\sqrt{58} > 5 + a\)
\(\sqrt{58} - 5 > a\)
Since \(\sqrt{58} \approx 7.615\), we have:
\(7.615 - 5 > a\)
\(2.615 > a\)
Also, the circles should not intersect, so:
C₁ C₂ > |r₁ - r₂|
\(\sqrt{58} > |5 - a|\)
\(-\sqrt{58} < 5 - a < \sqrt{58}\)
\(a - 5 < \sqrt{58}\) and \(5 - a < \sqrt{58}\)
\(a < 5 + \sqrt{58}\) and \(a > 5 - \sqrt{58}\)
\(a < 5 + 7.615\) and \(a > 5 - 7.615\)
\(a < 12.615\) and \(a > -2.615\)
Since a \(\in \mathbb{N}\), we have \(1 \le a \le 12\).
However, we need \(a < \sqrt{58} - 5\), so \(a < 2.615\).
Since a \(\in \mathbb{N}\), the only possible values are a = 1 and a = 2.
Thus, there are 2 possible values of a.
Therefore, the possible number of values of a is 2.