The number of direct common tangents that can be drawn to the circles `x^(2)+y^(2)+4x-6y-12=0 and x^(2)+y^(2)-8x+10y+16=0` is/are

Options

A. 3
B. 1
C. 2
D. 4

Answer

To find the number of direct common tangents that can be drawn to the given circles, we will follow these steps:

### Step 1: Write the equations of the circles in standard form
The given equations of the circles are:
1. \( x^2 + y^2 + 4x - 6y - 12 = 0 \)
2. \( x^2 + y^2 - 8x + 10y + 16 = 0 \)

We will rearrange these equations to identify the centers and radii.

### Step 2: Identify the center and radius of the first circle
For the first circle:
\[ x^2 + y^2 + 4x - 6y - 12 = 0 \]
We can rewrite this as:
\[ (x^2 + 4x) + (y^2 - 6y) = 12 \]
Completing the square:
\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 12 \]
\[ (x + 2)^2 + (y - 3)^2 = 25 \]
From this, we see that the center \(C_1\) is \((-2, 3)\) and the radius \(r_1\) is \(5\) (since \( \sqrt{25} = 5\)).

### Step 3: Identify the center and radius of the second circle
For the second circle:
\[ x^2 + y^2 - 8x + 10y + 16 = 0 \]
Rearranging gives:
\[ (x^2 - 8x) + (y^2 + 10y) = -16 \]
Completing the square:
\[ (x - 4)^2 - 16 + (y + 5)^2 - 25 = -16 \]
\[ (x - 4)^2 + (y + 5)^2 = 25 \]
Thus, the center \(C_2\) is \((4, -5)\) and the radius \(r_2\) is also \(5\).

### Step 4: Calculate the distance between the centers
Now we calculate the distance \(d\) between the centers \(C_1\) and \(C_2\):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(4 - (-2))^2 + (-5 - 3)^2} \]
\[ = \sqrt{(4 + 2)^2 + (-8)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 5: Determine the relationship between the distance and the radii
We have:
- \(d = 10\)
- \(r_1 + r_2 = 5 + 5 = 10\)

Since \(d = r_1 + r_2\), the circles are externally tangent to each other.

### Step 6: Conclusion about the number of direct common tangents
When two circles are externally tangent, there are exactly \(2\) direct common tangents.

Thus, the number of direct common tangents that can be drawn to the circles is **2**.

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