Let `C : x^2 + y^2 = 4` and `C’ : x^2 + y^2- 4 lambda x + 9 = 0` be two circles. If the set of all values of `lambda` so that the circles C and C’ intersect at two distinct points, is `R– [a, b]`, then the point `(8a + 12, 16b – 20)` lies on the curve :

Options

A. `x^2 + 2y^2 – 5x + 6y = 3`
B. `5x^2 – y = – 11`
C. `x^2 – 4y^2 = 7`
D. `6x^2 + y^2 = 42`

Answer

To solve the problem, we need to analyze the two circles given by their equations and find the conditions under which they intersect at two distinct points.

### Step 1: Identify the equations of the circles

The first circle \( C \) is given by:
\[ x^2 + y^2 = 4 \]
This is a circle with center at \( (0, 0) \) and radius \( r_1 = 2 \).

The second circle \( C' \) is given by:
\[ x^2 + y^2 - 4\lambda x + 9 = 0 \]
We can rearrange this to find its center and radius.

### Step 2: Rewrite the second circle in standard form

The equation of the second circle can be rewritten as:
\[ x^2 - 4\lambda x + y^2 + 9 = 0 \]
Completing the square for the \( x \) terms:
\[ (x^2 - 4\lambda x + 4\lambda^2) + y^2 = 4\lambda^2 - 9 \]
Thus, the equation becomes:
\[ (x - 2\lambda)^2 + y^2 = 4\lambda^2 - 9 \]
From this, we see that the center of circle \( C' \) is \( (2\lambda, 0) \) and its radius \( r_2 \) is:
\[ r_2 = \sqrt{4\lambda^2 - 9} \]

### Step 3: Find the condition for intersection

For the circles to intersect at two distinct points, the distance between their centers must be less than the sum of their radii and greater than the absolute difference of their radii:
\[ |r_1 - r_2| < d < r_1 + r_2 \]
Where \( d \) is the distance between the centers of the circles. The distance \( d \) between the centers \( (0, 0) \) and \( (2\lambda, 0) \) is:
\[ d = |2\lambda - 0| = 2|\lambda| \]

### Step 4: Set up the inequalities

1. **Absolute difference condition**:
\[ |2 - \sqrt{4\lambda^2 - 9}| < 2|\lambda| \]
2. **Sum condition**:
\[ 2 + \sqrt{4\lambda^2 - 9} > 2|\lambda| \]

### Step 5: Solve the inequalities

#### For the absolute difference condition:
1. Case 1: \( 2 - \sqrt{4\lambda^2 - 9} < 2|\lambda| \)
\[ \sqrt{4\lambda^2 - 9} > 2 - 2|\lambda| \]
Squaring both sides:
\[ 4\lambda^2 - 9 > (2 - 2|\lambda|)^2 \]
Expanding and simplifying gives:
\[ 4\lambda^2 - 9 > 4 - 8|\lambda| + 4\lambda^2 \]
Thus:
\[ -9 > 4 - 8|\lambda| \implies 8|\lambda| > 13 \implies |\lambda| > \frac{13}{8} \]

2. Case 2: \( \sqrt{4\lambda^2 - 9} - 2 < 2|\lambda| \)
\[ \sqrt{4\lambda^2 - 9} < 2 + 2|\lambda| \]
Squaring both sides:
\[ 4\lambda^2 - 9 < (2 + 2|\lambda|)^2 \]
Expanding gives:
\[ 4\lambda^2 - 9 < 4 + 8|\lambda| + 4\lambda^2 \]
Thus:
\[ -9 < 4 + 8|\lambda| \implies 8|\lambda| > -13 \text{ (always true)} \]

#### For the sum condition:
\[ 2 + \sqrt{4\lambda^2 - 9} > 2|\lambda| \]
This leads to:
\[ \sqrt{4\lambda^2 - 9} > 2|\lambda| - 2 \]
Squaring gives:
\[ 4\lambda^2 - 9 > (2|\lambda| - 2)^2 \]
Expanding and simplifying leads to:
\[ 4\lambda^2 - 9 > 4\lambda^2 - 8|\lambda| + 4 \implies -9 > -8|\lambda| + 4 \implies 8|\lambda| > 13 \implies |\lambda| > \frac{13}{8} \]

### Step 6: Combine results

The values of \( \lambda \) that satisfy both conditions are:
\[ \lambda \in (-\infty, -\frac{13}{8}) \cup (\frac{13}{8}, \infty) \]
Thus, \( a = -\frac{13}{8} \) and \( b = \frac{13}{8} \).

### Step 7: Calculate the point

Now, we need to find the point \( (8a + 12, 16b - 20) \):
1. Calculate \( 8a + 12 \):
\[ 8(-\frac{13}{8}) + 12 = -13 + 12 = -1 \]
2. Calculate \( 16b - 20 \):
\[ 16(\frac{13}{8}) - 20 = 26 - 20 = 6 \]

Thus, the point is \( (-1, 6) \).

### Step 8: Determine the curve

We need to check if the point \( (-1, 6) \) lies on the curve \( x^2 + y^2 = 4 \):
\[ (-1)^2 + 6^2 = 1 + 36 = 37 \neq 4 \]
So, it does not lie on the first circle.

Next, check if it lies on the second circle:
\[ (-1)^2 + 6^2 - 4\lambda(-1) + 9 = 0 \]
Substituting \( \lambda = \frac{13}{8} \):
\[ 1 + 36 + \frac{52}{8} + 9 = 0 \]
This is not valid.

### Conclusion

The point \( (-1, 6) \) does not lie on either of the circles.