An ellipse \(E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) passes through the vertices of the hyperbola \(H:\frac{x^2}{49} - \frac{y^2}{64} = -1\) Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.
The Vertices of hyperbola = (0, ± 8) As ellipse pass through it i.e., \(0+\frac{64}{b^2} = 1\) \(⇒ b^2 = 64...(1)\) As major axis of ellipse coincide with transverse axis of hyperbola we have b > a i.e. \(e_E = \sqrt{1 - \frac{a^2}{64}}\)\(= \frac{\sqrt{64-a^2}}{8}\) and \(e_H = \sqrt{1+\frac{49}{64}} = \frac{\sqrt{113}}{8}\) \(∴ e_E . e_H = \frac{1}{2}\frac{\sqrt{64-a^2}\sqrt{113}}{64}\) \(⇒ (64-a^2)(113) = 32^2\) \(⇒ a^2 = 64-\frac{1024}{113}\) L.R of ellipse \(= \frac{2a^2}{b}\) \(= \frac{2}{8}(\frac{113×64-1024}{113})\) \(I = \frac{1552}{113}\) \(∴ 113l = 1552\)
When a plane intersects a cone in multiple sections, several types of curves are obtained. These curves can be a circle, an ellipse, a parabola, and a hyperbola. When a plane cuts the cone other than the vertex then the following situations may occur:
Let ‘β’ is the angle made by the plane with the vertical axis of the cone
When β = 90°, we say the section is a circle
When α < β < 90°, then the section is an ellipse
When α = β; then the section is said to as a parabola
When 0 ≤ β < α; then the section is said to as a hyperbola
