The length of the tangent from a point on the circle `x^(2)+y^(2)+4x-6y-12=0` to the circle `x^(2)+y^(2)+4x-6y+4=0` is

Options

A. `2`
B. `16`
C. `8`
D. `4`

Answer

To find the length of the tangent from a point on the first circle to the second circle, we will follow these steps:

### Step 1: Write down the equations of the circles
The equations of the circles are given as:
1. Circle 1: \( x^2 + y^2 + 4x - 6y - 12 = 0 \)
2. Circle 2: \( x^2 + y^2 + 4x - 6y + 4 = 0 \)

### Step 2: Rearrange the equations to standard form
We can rearrange the equations to identify the center and radius of each circle.

For Circle 1:
\[ x^2 + y^2 + 4x - 6y - 12 = 0 \implies (x^2 + 4x) + (y^2 - 6y) = 12 \]
Completing the square:
\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 12 \implies (x + 2)^2 + (y - 3)^2 = 25 \]
Thus, the center is \((-2, 3)\) and the radius \(r_1 = \sqrt{25} = 5\).

For Circle 2:
\[ x^2 + y^2 + 4x - 6y + 4 = 0 \implies (x^2 + 4x) + (y^2 - 6y) = -4 \]
Completing the square:
\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = -4 \implies (x + 2)^2 + (y - 3)^2 = 9 \]
Thus, the center is also \((-2, 3)\) and the radius \(r_2 = \sqrt{9} = 3\).

### Step 3: Calculate the length of the tangent
The length of the tangent from a point on the first circle to the second circle can be calculated using the formula:
\[ \text{Length of tangent} = \sqrt{r_1^2 - r_2^2} \]
Substituting the values of \(r_1\) and \(r_2\):
\[ \text{Length of tangent} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \]

### Conclusion
The length of the tangent from a point on the first circle to the second circle is \(4\).

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