Key Concepts and Formulas

• Equation of a Circle: The general equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Condition for Intersection of Two Circles: Two circles with radii $r_1$ and $r_2$ and distance $d$ between their centers intersect at two distinct points if $|r_1 - r_2| < d < r_1 + r_2$.

Step-by-Step Solution

Step 1: Identify the centers and radii of the given circles.

The equation of circle $C$ is $x^2 + y^2 = 4$. Comparing this with the standard form $(x-h)^2 + (y-k)^2 = r^2$, we have center $C_1 = (0, 0)$ and radius $r_1 = \sqrt{4} = 2$.

The equation of circle $C'$ is $x^2 + y^2 - 4\lambda x + 9 = 0$. We can rewrite this equation by completing the square for the $x$ terms: $(x^2 - 4\lambda x) + y^2 + 9 = 0$ $(x^2 - 4\lambda x + (2\lambda)^2) + y^2 = (2\lambda)^2 - 9$ $(x - 2\lambda)^2 + y^2 = 4\lambda^2 - 9$ Comparing this with the standard form, we have center $C_2 = (2\lambda, 0)$ and radius $r_2 = \sqrt{4\lambda^2 - 9}$. For the radius to be real, we must have $4\lambda^2 - 9 > 0$, which means $\lambda^2 > \frac{9}{4}$, or $|\lambda| > \frac{3}{2}$.

Step 2: Calculate the distance between the centers of the circles.

The distance between the centers $C_1 = (0, 0)$ and $C_2 = (2\lambda, 0)$ is $d = \sqrt{(2\lambda - 0)^2 + (0 - 0)^2} = \sqrt{(2\lambda)^2} = |2\lambda| = 2|\lambda|$

Step 3: Apply the condition for the intersection of two circles.

For the circles to intersect at two distinct points, we need $|r_1 - r_2| < d < r_1 + r_2$, which means $|2 - \sqrt{4\lambda^2 - 9}| < 2|\lambda| < 2 + \sqrt{4\lambda^2 - 9}$

We have two inequalities to consider:

1. $2|\lambda| < 2 + \sqrt{4\lambda^2 - 9}$ $2|\lambda| - 2 < \sqrt{4\lambda^2 - 9}$ Since $|\lambda| > \frac{3}{2}$, $2|\lambda| > 3$, so $2|\lambda| - 2 > 0$. Squaring both sides, $(2|\lambda| - 2)^2 < 4\lambda^2 - 9$ $4\lambda^2 - 8|\lambda| + 4 < 4\lambda^2 - 9$ $13 < 8|\lambda|$ $|\lambda| > \frac{13}{8}$

2. $|2 - \sqrt{4\lambda^2 - 9}| < 2|\lambda|$ This inequality is equivalent to $-2|\lambda| < 2 - \sqrt{4\lambda^2 - 9} < 2|\lambda|$. We can split this into two inequalities:

   • $-2|\lambda| < 2 - \sqrt{4\lambda^2 - 9}$ $\sqrt{4\lambda^2 - 9} < 2|\lambda| + 2$ Since both sides are positive, we can square both sides: $4\lambda^2 - 9 < 4\lambda^2 + 8|\lambda| + 4$ $-13 < 8|\lambda|$ $|\lambda| > -\frac{13}{8}$, which is always true since $|\lambda|$ is non-negative.
   • $2 - \sqrt{4\lambda^2 - 9} < 2|\lambda|$ $2 - 2|\lambda| < \sqrt{4\lambda^2 - 9}$ We already considered this inequality in the first inequality, so $|\lambda| > \frac{13}{8}$.

Since we also need $|\lambda| > \frac{3}{2}$, the intersection of $|\lambda| > \frac{3}{2}$ and $|\lambda| > \frac{13}{8}$ means $|\lambda| > \frac{13}{8}$ (since $\frac{13}{8} > \frac{3}{2} = \frac{12}{8}$).

Therefore, $\lambda > \frac{13}{8}$ or $\lambda < -\frac{13}{8}$. So $\lambda \in (-\infty, -\frac{13}{8}) \cup (\frac{13}{8}, \infty)$. Thus, the set of values of $\lambda$ is $\mathbb{R} - [-\frac{13}{8}, \frac{13}{8}]$.

Step 4: Determine the values of a and b.

Comparing $\mathbb{R} - [-\frac{13}{8}, \frac{13}{8}]$ with $\mathbb{R} - [a, b]$, we have $a = -\frac{13}{8}$ and $b = \frac{13}{8}$.

Step 5: Calculate the coordinates of the point (8a + 12, 16b - 20).

$8a + 12 = 8(-\frac{13}{8}) + 12 = -13 + 12 = -1$ $16b - 20 = 16(\frac{13}{8}) - 20 = 2(13) - 20 = 26 - 20 = 6$ So the point is $(-1, 6)$.

Step 6: Check which curve the point (-1, 6) lies on.

(A) $x^2 + 2y^2 - 5x + 6y = 3$ $(-1)^2 + 2(6)^2 - 5(-1) + 6(6) = 1 + 2(36) + 5 + 36 = 1 + 72 + 5 + 36 = 114 \ne 3$ (Incorrect) $1 + 72 + 5 + 36 = 114$. It must equal 3.

Let's re-examine our solution. $|\lambda| > \frac{13}{8}$, which means $\lambda \in (-\infty, -\frac{13}{8}) \cup (\frac{13}{8}, \infty)$. So $a = -\frac{13}{8}$ and $b = \frac{13}{8}$.

$8a + 12 = 8(-\frac{13}{8}) + 12 = -13 + 12 = -1$ $16b - 20 = 16(\frac{13}{8}) - 20 = 2(13) - 20 = 26 - 20 = 6$ So the point is $(-1, 6)$.

(A) $x^2 + 2y^2 - 5x + 6y = 3$ $(-1)^2 + 2(6)^2 - 5(-1) + 6(6) = 1 + 72 + 5 + 36 = 114$. This is incorrect.

Let's double-check the inequalities. $2|\lambda| < 2 + \sqrt{4\lambda^2 - 9}$ $2|\lambda| - 2 < \sqrt{4\lambda^2 - 9}$ If $|\lambda| > 1$, we square both sides. $4\lambda^2 - 8|\lambda| + 4 < 4\lambda^2 - 9$ $13 < 8|\lambda|$ $|\lambda| > \frac{13}{8}$.

$|2 - \sqrt{4\lambda^2 - 9}| < 2|\lambda|$ $-2|\lambda| < 2 - \sqrt{4\lambda^2 - 9} < 2|\lambda|$ $\sqrt{4\lambda^2 - 9} < 2|\lambda| + 2$ and $2 - \sqrt{4\lambda^2 - 9} < 2|\lambda|$ $4\lambda^2 - 9 < 4\lambda^2 + 8|\lambda| + 4$, so $|\lambda| > -13/8$ (always true). $2 - 2|\lambda| < \sqrt{4\lambda^2 - 9}$ $4 - 8|\lambda| + 4\lambda^2 < 4\lambda^2 - 9$ $13 < 8|\lambda|$ $|\lambda| > \frac{13}{8}$.

Therefore, $\lambda \in (-\infty, -\frac{13}{8}) \cup (\frac{13}{8}, \infty)$, so $a = -\frac{13}{8}$ and $b = \frac{13}{8}$. The point is $(-1, 6)$.

(A) $x^2 + 2y^2 - 5x + 6y = 3$ $(-1)^2 + 2(6)^2 - 5(-1) + 6(6) = 1 + 72 + 5 + 36 = 114 \neq 3$

(B) $5x^2 - y = -11$ $5(-1)^2 - 6 = 5 - 6 = -1 \neq -11$

(C) $x^2 - 4y^2 = 7$ $(-1)^2 - 4(6)^2 = 1 - 4(36) = 1 - 144 = -143 \neq 7$

(D) $6x^2 + y^2 = 42$ $6(-1)^2 + (6)^2 = 6 + 36 = 42$

The point $(-1, 6)$ lies on $6x^2 + y^2 = 42$. However, the correct answer is (A). There must be an error in the provided answer key.

Common Mistakes & Tips

• Remember to consider both positive and negative roots when taking square roots, especially when dealing with absolute values.
• Always check the condition for the radius to be real ($r^2 > 0$).
• Be careful when squaring inequalities; make sure both sides are non-negative.

Summary

We determined the centers and radii of the two circles, applied the condition for the circles to intersect at two distinct points, and found the range of values for $\lambda$. Then, we calculated the coordinates of the point $(8a + 12, 16b - 20)$ and checked which curve it lies on. Based on our calculations, the point $(-1, 6)$ lies on the curve $6x^2 + y^2 = 42$, which corresponds to option (D). However, the given correct answer is option (A). There must be an error in the provided answer key.

Final Answer

The final answer is \boxed{D}, which contradicts the given answer (A).