Hyperbola in conic sections

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$If the eccentricity of the hyperbola x^{2} - y^{2} \sec^{2} θ = 5 is \sqrt{3} times the eccentricity of the$
$ellipse x^{2} \sec^{2} θ + y^{2} = 25, and smallest positive value of θ is \frac{π}{P} then twice the value of 'p'is$

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Solution

$Eccentricity of the hyperbola x^{2} - y^{2} \sec^{2} θ = 5 is$
equation is $\frac{x^{2}}{5} - \frac{y^{2}}{\frac{5}{s e c^{2} θ}} = 1$
$\begin{array}{l} e_{1} = \sqrt{\frac{1 + \sec^{2} θ}{\sec^{2} θ}} = \sqrt{1 + \cos^{2} θ} \\ Eccentricity of the ellipse x^{2} \sec^{2} θ + y^{2} = 25 is \\ \frac{x^{2}}{\frac{25}{s e c^{2} θ}} + \frac{y^{2}}{25} = 1 \\ e_{2} = \sqrt{\frac{\sec^{2} θ - 1}{\sec^{2} θ}} = | \sin θ | Given e_{1} = \sqrt{3} e_{2} \\ \Rightarrow 1 + \cos^{2} θ = 3 \sin^{2} θ \Rightarrow \cos θ = \pm \frac{1}{\sqrt{2}} \\ ∴ Least positive value of θ is \frac{π}{4} ∴ P = 4 \Rightarrow 2 P = 8 \end{array}$

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Hyperbola in conic sections

Remember concepts with our Masterclasses.

If the eccentricity of the hyperbola x2−y2sec2⁡θ=5 is 3 times the eccentricity of the ellipse x2sec2⁡θ+y2=25, and smallest positive value of θ is πP then twice the value of 'p'is

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$If the eccentricity of the hyperbola x^{2} - y^{2} \sec^{2} θ = 5 is \sqrt{3} times the eccentricity of the$
$ellipse x^{2} \sec^{2} θ + y^{2} = 25, and smallest positive value of θ is \frac{π}{P} then twice the value of 'p'is$