Two circles `c_1:x^2+y^2-4x-6y-3=0` and `c_2:x^2+y^2+2x-14y+lamda` meet at two distinct points then find the value of `lamda`.

Options

A. `-31ltlamdalt40`
B. `-31ltlamdalt49`
C. `-20ltlamdalt49`
D. `-11ltlamdalt49`

Answer

To solve the problem of finding the value of \(\lambda\) such that the circles \(c_1: x^2 + y^2 - 4x - 6y - 3 = 0\) and \(c_2: x^2 + y^2 + 2x - 14y + \lambda = 0\) meet at two distinct points, we will follow these steps:

### Step 1: Rewrite the equations of the circles in standard form

**Circle \(c_1\):**
The equation is given by:
\[ x^2 + y^2 - 4x - 6y - 3 = 0 \]
We can rearrange it as:
\[ x^2 - 4x + y^2 - 6y = 3 \]
Now, we complete the square for \(x\) and \(y\).

For \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
For \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]
Substituting these back into the equation gives:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 = 3 \]
Simplifying this, we have:
\[ (x - 2)^2 + (y - 3)^2 = 16 \]
Thus, the center and radius of circle \(c_1\) are:
- Center \(C_1 = (2, 3)\)
- Radius \(r_1 = 4\)

**Circle \(c_2\):**
The equation is given by:
\[ x^2 + y^2 + 2x - 14y + \lambda = 0 \]
Rearranging gives:
\[ x^2 + 2x + y^2 - 14y = -\lambda \]
Completing the square for \(x\) and \(y\):

For \(x\):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
For \(y\):
\[ y^2 - 14y = (y - 7)^2 - 49 \]
Substituting these back gives:
\[ (x + 1)^2 - 1 + (y - 7)^2 - 49 = -\lambda \]
This simplifies to:
\[ (x + 1)^2 + (y - 7)^2 = 50 - \lambda \]
Thus, the center and radius of circle \(c_2\) are:
- Center \(C_2 = (-1, 7)\)
- Radius \(r_2 = \sqrt{50 - \lambda}\)

### Step 2: Find the distance between the centers of the circles

The distance \(d\) between the centers \(C_1\) and \(C_2\) is calculated as:
\[ d = \sqrt{(2 - (-1))^2 + (3 - 7)^2} = \sqrt{(2 + 1)^2 + (3 - 7)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Apply the condition for two distinct intersection points

For the circles to intersect at two distinct points, the following condition must hold:
\[ |r_1 - r_2| < d < r_1 + r_2 \]
Substituting the values:
1. \(r_1 - r_2 = 4 - \sqrt{50 - \lambda}\)
2. \(r_1 + r_2 = 4 + \sqrt{50 - \lambda}\)

Thus, the inequalities become:
\[ |4 - \sqrt{50 - \lambda}| < 5 < 4 + \sqrt{50 - \lambda} \]

### Step 4: Solve the inequalities

**First Inequality:**
\[ 4 - \sqrt{50 - \lambda} < 5 \implies -\sqrt{50 - \lambda} < 1 \implies \sqrt{50 - \lambda} > -1 \quad \text{(always true)} \]

Now for the other part:
\[ \sqrt{50 - \lambda} < 9 \implies 50 - \lambda < 81 \implies \lambda > -31 \]

**Second Inequality:**
\[ 4 + \sqrt{50 - \lambda} > 5 \implies \sqrt{50 - \lambda} > 1 \implies 50 - \lambda > 1 \implies \lambda < 49 \]

### Step 5: Combine the results

From the inequalities, we have:
\[ -31 < \lambda < 49 \]

Thus, the value of \(\lambda\) for which the circles \(c_1\) and \(c_2\) meet at two distinct points is:
\[ \lambda \in (-31, 49) \]