Answer the following by appropriately matching the lists based on the information given in the paragraph Let the circles C1 : x2 + y2 = 9 and C2 : (x – 3)2 + (y – 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x – h)2 + (y – k)2 = r2 satisfies the following conditions : (i) centre of C3 is collinear with the centres of C1 and C2, (ii) C1 and C2 both lie inside C3, and (iii) C3 touches C1 at M and C2 at N. Let the line through X and Y intersect C3 at Z and W, and let a common tangent of C1 and C3 be a tangent to the parabola x2 = 8αy. There are some expression given in the List-I whose values are given in List-II below: List - I List - II (I) 2h + k (P) 6 (II) \frac{Length \textrm{ }\textrm{ } of \textrm{ }\textrm{ } ZW}{Length \textrm{ } of \textrm{ } XY} (Q) \sqrt{6} (III) \frac{Area \textrm{ } of \textrm{ } triangle \textrm{ } MZN}{Area \textrm{ } of \textrm{ } triangle \textrm{ } ZMW} (R) \frac{5}{4} (IV) α (S) \frac{21}{5} (T) 2 \sqrt{6} (U) \frac{10}{3} Which of the following is the only INCORRECT combination ?

Engineering

Mathematics

Common Tangent To two Circle

Question

Answer the following by appropriately matching the lists based on the information given in the paragraph

Let the circles C₁ : x² + y² = 9 and C₂ : (x – 3)² + (y – 4)² = 16, intersect at the points X and Y.

Suppose that another circle C₃ : (x – h)² + (y – k)² = r² satisfies the following conditions :

(i) centre of C₃ is collinear with the centres of C₁ and C₂,

(ii) C₁ and C₂ both lie inside C₃, and

(iii) C₃ touches C₁ at M and C₂ at N.

Let the line through X and Y intersect C₃ at Z and W, and let a common tangent of C₁ and C₃ be a tangent to the parabola x² = 8αy.

There are some expression given in the List-I whose values are given in List-II below:

List - I	List - II
(I) 2h + k	(P) 6
(II) $\frac{Length of ZW}{Length of XY}$	(Q) $\sqrt{6}$
(III) $\frac{Area of triangle MZN}{Area of triangle ZMW}$	(R) $\frac{5}{4}$
(IV) α	(S) $\frac{21}{5}$
	(T) $2 \sqrt{6}$
	(U) $\frac{10}{3}$

Which of the following is the only INCORRECT combination ?

(IV), (U)

(III), (R)

(I), (P)

(IV), (S)

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MC₁ + C₁C₂ + C₂N = 2r

⇒ 3 + 5 + 4 = 2r ⇒ r = 6 ⇒ Radius of C₃ = 6

Suppose centre of C₃ be (0 + r₄ cos θ, 0 + r₄ sin θ), ${\begin{matrix} r_{4} = C_{1} C_{3} = 3 \\ tanθ = \frac{4}{3} \end{matrix}}$

C₃ = $(\frac{9}{5}, \frac{12}{5})$ = (h, k) ⇒ 2h + k = 6

Equation of ZW and XY is 3x + 4y – 9 = 0

(common chord of circle C₁ = 0 and C₂ = 0)

$ZW = \sqrt{r^{2} - p^{2}} = \frac{24 \sqrt{6}}{5} (where r = 6 and p = \frac{6}{5})$

$XY = 2 \sqrt{r_{1}^{2} - p_{1}^{2}} = \frac{24}{5} (where r_{1} = 3 and p_{1} = \frac{9}{5})$

$\frac{Length of ZW}{Length of XY} = \sqrt{6}$

Let length of perpendicular from M to ZW be $λ, λ = 3 + \frac{9}{5} = \frac{24}{5}$

$= \frac{Area of ΔMZN}{Area of ΔZMW} = \frac{\frac{1}{2} (MN) \times \frac{1}{2} (ZW)}{\frac{1}{2} \times ZW \times λ} = \frac{MN}{2 λ} = \frac{5}{4}$

$C_{3} : {(x - \frac{9}{5})}^{2} + {(y - \frac{12}{5})}^{2} = 6^{2}$

C₁ : x² + y² – 9 = 0

common tangent to C₁ and C₃ is common chord of C₁ and C₃ is 3x + 4y + 15 = 0.

Now, 3x + 4y + 15 = 0 is tangent to parabola x² = 8αy.

x² = 8α $(\frac{- 3 x - 15}{4})$ ⇒ 4x² + 24αx + 120α = 0

D = 0 ⇒ $α = \frac{10}{3} .$