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(B) The equation of the hyperbola is $\frac{x^2}{9} - \frac{y^2}{4} = 1$. Let $P(3 \sec 	heta, 2 	an 	heta)$ be any point on the hyperbola.The equa

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(B) The equation of the hyperbola is $\frac{x^2}{9} - \frac{y^2}{4} = 1$. Let $P(3 \sec 	heta, 2 	an 	heta)$ be any point on the hyperbola.The equation of the chord of contact $AB$ with respect to $P$ for the circle $x^2 + y^2 = 9$ is $T = 0$,which is $(3 \sec 	heta)x + (2 	an 	heta)y = 9$ ..... $(1)$.Also,the equation of a chord with a given midpoint $(h, k)$ for the circle $x^2 + y^2 = 9$ is $T = S_1$,which is $hx + ky = h^2 + k^2$ or $hx + ky - (h^2 + k^2) = 0$ ..... $(2)$.Comparing $(1)$ and $(2)$,we have $\frac{3 \sec 	heta}{h} = \frac{2 	an 	heta}{k} = \frac{9}{h^2 + k^2}$.Thus,$\sec 	heta = \frac{3h}{h^2 + k^2}$ and $	an 	heta = \frac{9k}{2(h^2 + k^2)}$.Using the identity $\sec^2 	heta - 	an^2 	heta = 1$,we get $\left( \frac{3h}{h^2 + k^2} ight)^2 - \left( \frac{9k}{2(h^2 + k^2)} ight)^2 = 1$.$\frac{9h^2}{(h^2 + k^2)^2} - \frac{81k^2}{4(h^2 + k^2)^2} = 1$.Multiplying by $\frac{(h^2 + k^2)^2}{9}$,we get $h^2 - \frac{9k^2}{4} = \frac{(h^2 + k^2)^2}{9}$.Dividing by $9$,we get $\frac{h^2}{9} - \frac{k^2}{4} = \frac{(h^2 + k^2)^2}{81} = \left( \frac{h^2 + k^2}{9} ight)^2$.Replacing $(h, k)$ with $(x, y)$,the locus is $\left( \frac{x^2}{9} - \frac{y^2}{4} ight) = 1 \cdot \left( \frac{x^2 + y^2}{9} ight)^2$.Comparing this with the given equation,we find $\lambda = 1$.

Tangents are drawn from any point on the hyperbola $4x^2 - 9y^2 = 36$ to the circle $x^2 + y^2 = 9$. If the locus of the midpoint of the chord of contact is $\left( \frac{x^2}{9} - \frac{y^2}{4} \right) = \lambda \left( \frac{x^2 + y^2}{9} \right)^2$,then the value of $\lambda$ is:

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