Consider a hyperbola H having centre at the origin and foci on the x-axis. Let be the circle touching the hyperbola H and having the centre at the origin. Let be the circle touching the hyperbola H at its vertex and having the centre at one of its foci.

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Solution

Q.
Consider a hyperbola H having centre at the origin and foci on the x-axis. Let $C_{1}$ be the circle touching the hyperbola H and having the centre at the origin. Let $C_{2}$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq. units) of $C_{1}$ and $C_{2}$ are 36 $π$ and 4 $π$ , respectively, then the length (in units) of latus rectum of H is [2024]

1 $\frac{28}{3}$

2 $\frac{14}{3}$

3 $\frac{10}{3}$

4 $\frac{11}{3}$

Ans.
(1)

$C_{1} : x^{2} + y^{2} = a^{2}$

$\Rightarrow Area = {πa}^{2} = 36 π \Rightarrow a = 6$

$C_{2} : {(x - a e)}^{2} + y^{2} = {(a e - a)}^{2}$

$\Rightarrow Area = π {(a e - a)}^{2} = 4 π \Rightarrow 36 {(e - 1)}^{2} = 4$

$\Rightarrow e - 1 = \frac{1}{3} \Rightarrow e = \frac{4}{3} \Rightarrow b^{2} = 28$

$∴$ Length of Latus Rectum = $\frac{2 b^{2}}{a} = \frac{2 \times 28}{6} = \frac{28}{3}$ units.

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