The range of values of `lambda` for which the circles `x^(2)+y^(2)=4` and `x^(2)+y^(2)-4lambda x + 9 = 0` have two common tangents, is

Options

A. `lambda in [ - ( 13)/( 8) , (13)/( 8)]`
B. `lambda gt ( 13)/( 8) ` or `lambda lt - ( 13)/( 8)`
C. ` 1 ltlambda lt ( 13)/(8)`
D. None of these

Answer

To solve the problem of finding the range of values of \( \lambda \) for which the circles \( x^2 + y^2 = 4 \) and \( x^2 + y^2 - 4\lambda x + 9 = 0 \) have two common tangents, we will follow these steps:

### Step 1: Identify the centers and radii of the circles

The first circle is given by:
\[ x^2 + y^2 = 4 \]
This can be rewritten in standard form as:
\[ (x - 0)^2 + (y - 0)^2 = 2^2 \]
From this, we can see that:
- Center \( C_1 = (0, 0) \)
- Radius \( r_1 = 2 \)

The second circle is given by:
\[ x^2 + y^2 - 4\lambda x + 9 = 0 \]
Rearranging gives:
\[ x^2 - 4\lambda x + y^2 + 9 = 0 \]
Completing the square for the \( x \) terms:
\[ (x - 2\lambda)^2 + y^2 = 4\lambda^2 - 9 \]
From this, we can see that:
- Center \( C_2 = (2\lambda, 0) \)
- Radius \( r_2 = \sqrt{4\lambda^2 - 9} \)

### Step 2: Set up the conditions for two common tangents

For two circles to have two common tangents, the distance between their centers \( C_1C_2 \) must satisfy the following condition:
\[ |C_1C_2| < r_1 + r_2 \quad \text{and} \quad |C_1C_2| > |r_1 - r_2| \]

### Step 3: Calculate the distance between the centers

The distance \( C_1C_2 \) is:
\[ C_1C_2 = |C_1C_2| = |2\lambda - 0| = 2\lambda \]

### Step 4: Apply the first condition

Using the first condition \( C_1C_2 < r_1 + r_2 \):
\[ 2\lambda < 2 + \sqrt{4\lambda^2 - 9} \]
Squaring both sides:
\[ (2\lambda)^2 < (2 + \sqrt{4\lambda^2 - 9})^2 \]
\[ 4\lambda^2 < 4 + 4\sqrt{4\lambda^2 - 9} + (4\lambda^2 - 9) \]
\[ 4\lambda^2 < 4 + 4\sqrt{4\lambda^2 - 9} + 4\lambda^2 - 9 \]
\[ 0 < 4 + 4\sqrt{4\lambda^2 - 9} - 9 \]
\[ 9 > 4\sqrt{4\lambda^2 - 9} \]
Dividing by 4:
\[ \frac{9}{4} > \sqrt{4\lambda^2 - 9} \]
Squaring both sides:
\[ \left(\frac{9}{4}\right)^2 > 4\lambda^2 - 9 \]
\[ \frac{81}{16} > 4\lambda^2 - 9 \]
Adding 9 to both sides:
\[ \frac{81}{16} + 9 > 4\lambda^2 \]
Converting 9 to sixteenths:
\[ \frac{81}{16} + \frac{144}{16} > 4\lambda^2 \]
\[ \frac{225}{16} > 4\lambda^2 \]
Dividing by 4:
\[ \frac{225}{64} > \lambda^2 \]
Taking square roots:
\[ \lambda < \frac{15}{8} \quad \text{and} \quad \lambda > -\frac{15}{8} \]

### Step 5: Apply the second condition

Using the second condition \( C_1C_2 > |r_1 - r_2| \):
\[ 2\lambda > |2 - \sqrt{4\lambda^2 - 9}| \]
This gives us two cases to consider:
1. \( 2\lambda > 2 - \sqrt{4\lambda^2 - 9} \)
2. \( 2\lambda > \sqrt{4\lambda^2 - 9} - 2 \)

For case 1:
\[ 2\lambda - 2 > -\sqrt{4\lambda^2 - 9} \]
Squaring both sides:
\[ (2\lambda - 2)^2 > 4\lambda^2 - 9 \]
Expanding:
\[ 4\lambda^2 - 8\lambda + 4 > 4\lambda^2 - 9 \]
\[ -8\lambda + 4 > -9 \]
\[ -8\lambda > -13 \]
\[ \lambda < \frac{13}{8} \]

For case 2:
\[ 2\lambda + 2 > \sqrt{4\lambda^2 - 9} \]
Squaring both sides:
\[ (2\lambda + 2)^2 > 4\lambda^2 - 9 \]
Expanding:
\[ 4\lambda^2 + 8\lambda + 4 > 4\lambda^2 - 9 \]
\[ 8\lambda + 4 > -9 \]
\[ 8\lambda > -13 \]
\[ \lambda > -\frac{13}{8} \]

### Step 6: Combine the results

From the first condition, we found:
\[ -\frac{15}{8} < \lambda < \frac{15}{8} \]
From the second condition, we found:
\[ -\frac{13}{8} < \lambda < \frac{13}{8} \]

Combining these inequalities, we find that the range of \( \lambda \) for which the circles have two common tangents is:
\[ \lambda < -\frac{15}{8} \quad \text{or} \quad \lambda > \frac{15}{8} \]

### Final Answer:
The range of values of \( \lambda \) for which the circles have two common tangents is:
\[ \lambda < -\frac{15}{8} \quad \text{or} \quad \lambda > \frac{15}{8} \]