If AB is a double ordinate of the hyperbola (x^2)/(a^2)-(y^2)/( b^2)=1 such that OAB is an equil

step 1 Hello students in this question it is given that AB is the double ordinate of this hyperbola and

If AB is a double ordinate of the hyperbola...

If $\mathrm{AB}$ is a double ordinate of the hyperbola $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ such that $\triangle \mathrm{OAB}$ is an equilateral triangle, where $\mathrm{O}$ is the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies (a) $1<\mathrm{e}<\frac{2}{\sqrt{3}}$ (b) $\mathrm{e}<\frac{1}{\sqrt{3}}$ (c) e $>\frac{2}{\sqrt{3}}$ (d) $\mathrm{e}=\frac{2}{\sqrt{3}}$

If $\mathrm{AB}$ is a double ordinate of the hyperbola $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ such that $\triangle \mathrm{OAB}$ is an equilateral triangle, where $\mathrm{O}$ is the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies
(a) $1<\mathrm{e}<\frac{2}{\sqrt{3}}$
(b) $\mathrm{e}<\frac{1}{\sqrt{3}}$
(c) e $>\frac{2}{\sqrt{3}}$
(d) $\mathrm{e}=\frac{2}{\sqrt{3}}$

Key Concepts

Relationship Between Conic Parameters

When a double ordinate of a hyperbola is involved in forming an equilateral triangle with the center, the geometric conditions combine with the hyperbola's equation to set up equations relating the coordinates of the endpoints and the parameters a and b. This often leads to a specific range or value for the eccentricity, since eccentricity is directly linked to a and b by the relation e = ?(1 + (b²/a²)).

Hyperbola

A hyperbola is a conic section defined by a quadratic equation in two variables, typically written in standard form as x²/a² - y²/b² = 1. Its graph consists of two separate branches, and it is characterized by its transverse and conjugate axes, center, and asymptotes.

Eccentricity

The eccentricity of a hyperbola is a measure of its deviation from being circular and is defined by the formula e = ?(1 + (b²/a²)) for a hyperbola in standard form. It is always greater than 1, and a larger eccentricity indicates a more stretched shape.

Double Ordinate

In the context of a hyperbola, a double ordinate refers to a chord drawn perpendicular to the transverse axis (usually parallel to the conjugate axis). This chord has endpoints that are symmetric about the center of the hyperbola, and when it is used in problems, its properties help relate the coordinates of the endpoints to the parameters of the hyperbola.

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are of equal length. In problems involving conic sections, requiring a triangle (formed by specific points such as the center and endpoints of a chord) to be equilateral introduces additional geometric constraints that enable relationships to be set up between the conic parameters.

Transcript

Please give Ace some feedback