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

### Related Formula For two circles with centers C_1, C_2$C_1, C_2$ and radii r_1, r_2$r_1, r_2$ to intersect at two distinct real points, the distance between their centers must satisfy: |r_1 - r_2| < C_1C_2 < r_1 + r_2$$|r_1 - r_2| < C_1C_2 < r_1 + r_2$$ ### Core Logic Extracting properties from the circle equations: - Circle C$C$: x^2 + y^2 = 4 implies$x^2 + y^2 = 4 \implies$ Center C_1 = (0,0)$C_1 = (0,0)$, Radius r_1 = 2$r_1 = 2$ - Circle C'$C'$: x^2 + y^2 - 4lambda x + 9 = 0 implies$x^2 + y^2 - 4\lambda x + 9 = 0 \implies$ Center C_2 = (2lambda, 0)$C_2 = (2\lambda, 0)$, Radius r_2 = sqrt(-2lambda)^2 - 9 = sqrt4lambda^2 - 9$r_2 = \sqrt{(-2\lambda)^2 - 9} = \sqrt{4\lambda^2 - 9}$ For real intersection conditions, the radius must be defined: 4lambda^2 - 9 > 0 implies lambda^2 > frac94 quad implies (1)$$4\lambda^2 - 9 > 0 \implies \lambda^2 > \frac{9}{4} \quad \implies (1)$$ Distance between centers: C_1C_2 = sqrt(2lambda - 0)^2 + 0^2 = |2lambda|$C_1C_2 = \sqrt{(2\lambda - 0)^2 + 0^2} = |2\lambda|$. ### Step 1: Formulate the Triangle Inequality Solutions Applying the intersection condition: |2 - sqrt4lambda^2 - 9| < |2lambda| < 2 + sqrt4lambda^2 - 9$$|2 - \sqrt{4\lambda^2 - 9}| < |2\lambda| < 2 + \sqrt{4\lambda^2 - 9}$$ The right side inequality |2lambda| < 2 + sqrt4lambda^2 - 9$|2\lambda| < 2 + \sqrt{4\lambda^2 - 9}$ is always valid for real radii. Solving the left side inequality by squaring: left(2 - sqrt4lambda^2 - 9right)^2 < (2lambda)^2$$\left(2 - \sqrt{4\lambda^2 - 9}\right)^2 < (2\lambda)^2$$ 4 + (4lambda^2 - 9) - 4sqrt4lambda^2 - 9 < 4lambda^2$$4 + (4\lambda^2 - 9) - 4\sqrt{4\lambda^2 - 9} < 4\lambda^2$$ -5 - 4sqrt4lambda^2 - 9 < 0 implies 4sqrt4lambda^2 - 9 > -5$$-5 - 4\sqrt{4\lambda^2 - 9} < 0 \implies 4\sqrt{4\lambda^2 - 9} > -5$$ Since a square root is always non-negative, square both sides to get the strict bound: 16(4lambda^2 - 9) > 25 implies 64lambda^2 - 144 > 25$$16(4\lambda^2 - 9) > 25 \implies 64\lambda^2 - 144 > 25$$ 64lambda^2 > 169 implies lambda^2 > frac16964 quad implies (2)$$64\lambda^2 > 169 \implies \lambda^2 > \frac{169}{64} \quad \implies (2)$$ Combining bounds (1) and (2), condition (2) is more restrictive, meaning: lambda in left(-infty, -frac138right) cup left(frac138, inftyright)$$\lambda \in \left(-\infty, -\frac{13}{8}\right) \cup \left(\frac{13}{8}, \infty\right)$$ This can be written as mathbbR - left[-frac138, frac138right]$\mathbb{R} - \left[-\frac{13}{8}, \frac{13}{8}\right]$. ### Step 2: Coordinate Analysis of the Target Point Comparing our interval with mathbbR - [a, b]$\mathbb{R} - [a, b]$, we identify: a = -frac138, quad b = frac138$$a = -\frac{13}{8}, \quad b = \frac{13}{8}$$ Now, substitute these bounds to locate the coordinates of our target point (8a+12, \, 16b-20)$(8a+12, \, 16b-20)$: - x$x$-coordinate: 8left(-frac138right) + 12 = -13 + 12 = -1$8\left(-\frac{13}{8}\right) + 12 = -13 + 12 = -1$ - y$y$-coordinate: 16left(frac138right) - 20 = 26 - 20 = 6$16\left(\frac{13}{8}\right) - 20 = 26 - 20 = 6$ Hence, the point is (-1, 6)$(-1, 6)$. ### Step 3: Test Point against the Options Substitute (-1, 6)$(-1, 6)$ into the given curves to find a match: Testing Option (4): 6x^2 + y^2 = 42$6x^2 + y^2 = 42$ 6(-1)^2 + (6)^2 = 6(1) + 36 = 42$$6(-1)^2 + (6)^2 = 6(1) + 36 = 42$$ Since LHS = RHS, the point satisfies the curve in Option (4). ### Pattern Recognition Sees: Dynamic variable interval limits forming loci requirements. Shortcut: When checking inequalities like |2 - sqrtz| < 2lambda$|2 - \sqrt{z}| < 2\lambda$, algebraic squaring helps simplify the terms quickly by eliminating the parameter 4lambda^2$4\lambda^2$ from both sides. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Conic Sections (Circles) Class 11 Mathematics: Linear Inequalities